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Ingy döt Net
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(phixonline)-->
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<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
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<span style="color: #000080;font-style:italic;">--
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-- While a phix dictionary can handle keys of {x,a}, for this
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-- task performance was dreadful (7,612,479 entries, >3mins),
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-- so instead memophix maps known x to an index to memophia
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-- which holds the full [1..a] for each x, dropping to a much
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-- more respectable (albeit not super-fast) 14s
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--</span>
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<span style="color: #008080;">constant</span> <span style="color: #000000;">memophix</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">new_dict</span><span style="color: #0000FF;">()</span>
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<span style="color: #004080;">sequence</span> <span style="color: #000000;">memophia</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span> <span style="color: #000080;font-style:italic;">-- 1..a (max 3401) for each x</span>
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<span style="color: #008080;">function</span> <span style="color: #000000;">phi</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
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<span style="color: #008080;">if</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">x</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
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<span style="color: #004080;">integer</span> <span style="color: #000000;">adx</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">getd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">memophix</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">res</span>
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<span style="color: #008080;">if</span> <span style="color: #000000;">adx</span><span style="color: #0000FF;">=</span><span style="color: #004600;">NULL</span> <span style="color: #008080;">then</span>
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<span style="color: #000000;">memophia</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">append</span><span style="color: #0000FF;">(</span><span style="color: #000000;">memophia</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">))</span>
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<span style="color: #000000;">adx</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">memophia</span><span style="color: #0000FF;">)</span>
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<span style="color: #7060A8;">setd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">adx</span><span style="color: #0000FF;">,</span><span style="color: #000000;">memophix</span><span style="color: #0000FF;">)</span>
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<span style="color: #008080;">else</span>
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<span style="color: #004080;">object</span> <span style="color: #000000;">ma</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">memophia</span><span style="color: #0000FF;">[</span><span style="color: #000000;">adx</span><span style="color: #0000FF;">]</span>
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<span style="color: #004080;">integer</span> <span style="color: #000000;">l</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">ma</span><span style="color: #0000FF;">)</span>
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<span style="color: #008080;">if</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">></span><span style="color: #000000;">l</span> <span style="color: #008080;">then</span>
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<span style="color: #000000;">memophia</span><span style="color: #0000FF;">[</span><span style="color: #000000;">adx</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span> <span style="color: #000080;font-style:italic;">-- kill refcount</span>
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<span style="color: #000000;">memophia</span><span style="color: #0000FF;">[</span><span style="color: #000000;">adx</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">ma</span> <span style="color: #0000FF;">&</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">-</span><span style="color: #000000;">l</span><span style="color: #0000FF;">)</span>
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<span style="color: #008080;">else</span>
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<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">ma</span><span style="color: #0000FF;">[</span><span style="color: #000000;">a</span><span style="color: #0000FF;">]</span>
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<span style="color: #008080;">if</span> <span style="color: #000000;">res</span><span style="color: #0000FF;">>=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">res</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
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<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
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<span style="color: #000000;">ma</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span> <span style="color: #000080;font-style:italic;">-- kill refcount</span>
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<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
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<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">phi</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">phi</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">/</span><span style="color: #7060A8;">get_prime</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)),</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
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<span style="color: #000000;">memophia</span><span style="color: #0000FF;">[</span><span style="color: #000000;">adx</span><span style="color: #0000FF;">][</span><span style="color: #000000;">a</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">res</span>
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<span style="color: #008080;">return</span> <span style="color: #000000;">res</span>
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<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
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<span style="color: #008080;">function</span> <span style="color: #000000;">pi</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
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<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;"><</span><span style="color: #000000;">2</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
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<span style="color: #004080;">integer</span> <span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">pi</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)))</span>
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<span style="color: #008080;">return</span> <span style="color: #000000;">phi</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">a</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">1</span>
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<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
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<span style="color: #004080;">atom</span> <span style="color: #000000;">t0</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()</span>
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<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">to</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">platform</span><span style="color: #0000FF;">()=</span><span style="color: #004600;">JS</span><span style="color: #0000FF;">?</span><span style="color: #000000;">8</span><span style="color: #0000FF;">:</span><span style="color: #000000;">9</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
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<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"10^%d %d\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">i</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pi</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">10</span><span style="color: #0000FF;">,</span><span style="color: #000000;">i</span><span style="color: #0000FF;">))})</span>
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<span style="color: #000080;font-style:italic;">-- printf(1,"10^%d %d\n",{i,length(get_primes_le(power(10,i)))})</span>
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
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<span style="color: #0000FF;">?</span><span style="color: #7060A8;">elapsed</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">time</span><span style="color: #0000FF;">()-</span><span style="color: #000000;">t0</span><span style="color: #0000FF;">)</span>
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<!--
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@ -0,0 +1,139 @@
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(phixonline)-->
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<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
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<span style="color: #7060A8;">requires</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"1.0.2"</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- (for in, tagstart)</span>
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<span style="color: #008080;">function</span> <span style="color: #000000;">half</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">return</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">((</span><span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)+</span><span style="color: #000000;">1</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span> <span style="color: #000080;font-style:italic;">// convenience convert to idx</span>
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<span style="color: #008080;">function</span> <span style="color: #000000;">count_primes</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
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<span style="color: #000080;font-style:italic;">// non-recursive Legendre prime counting function for a range `n`...
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// has O(n^(3/4)/((log n)^2)) time complexity; O(n^(1/2)) space complexity.</span>
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<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;"><</span><span style="color: #000000;">3</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #7060A8;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;"><</span><span style="color: #000000;">2</span><span style="color: #0000FF;">?</span><span style="color: #000000;">0</span><span style="color: #0000FF;">:</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #000080;font-style:italic;">// can't odd sieve for n less than 3!</span>
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<span style="color: #004080;">integer</span> <span style="color: #000000;">sqrtn</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">trunc</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)),</span> <span style="color: #000080;font-style:italic;">// (actual limit)</span>
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<span style="color: #000000;">mxndx</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">((</span><span style="color: #000000;">sqrtn</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">// odds-only limit
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--
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-- smalls is the current accumulated counts of odd primes 1 to sqrt(n), initialized
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-- to odds-only sieving, ie {0,1,2,3,4...} meaning 0 odd primes to 1, 1 o.p to 3,...
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--
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-- roughs is the current odd k-rough numbers up to sqrt of range; k = 2
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-- initialized to all odd positive numbers 1, 3, 5, 7, 9, 11, ... sqrt(n)
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--
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-- larges is an array of current phi counts for the above roughs... except they are
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-- not strictly `phi`'s since they also include primes, to match `smalls` above!
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-- initialized for current roughs after accounting for the even prime of two...
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--
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-- composite is a flag array representing odd numbers 1..sqrtn, for sieving.
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-- initialized false, meaning all positive odd numbers are potentially prime
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-- note that this array starts at (and keeps) 1 to match the algorithm even
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-- though 1 is not actually a prime, as 1 is important in computation of phi...
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--</span>
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<span style="color: #004080;">sequence</span> <span style="color: #000000;">smalls</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">tagset</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mxndx</span><span style="color: #0000FF;">,</span><span style="color: #000000;">0</span><span style="color: #0000FF;">),</span>
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<span style="color: #000000;">roughs</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">tagstart</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">mxndx</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">),</span>
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<span style="color: #000000;">larges</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sq_floor_div</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_div</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">roughs</span><span style="color: #0000FF;">),</span><span style="color: #000000;">1</span><span style="color: #0000FF;">),</span><span style="color: #000000;">2</span><span style="color: #0000FF;">),</span>
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<span style="color: #000000;">composite</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #004600;">false</span><span style="color: #0000FF;">,</span><span style="color: #000000;">mxndx</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
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<span style="color: #004080;">integer</span> <span style="color: #000000;">bp</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000080;font-style:italic;">// 'current' base prime</span>
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<span style="color: #000000;">nbp</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000080;font-style:italic;">// number of base primes found </span>
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<span style="color: #000000;">mxri</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">mxndx</span><span style="color: #0000FF;">,</span> <span style="color: #000080;font-style:italic;">// current highest used rough index</span>
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<span style="color: #000000;">i</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">sqri</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">4</span> <span style="color: #000080;font-style:italic;">// index and square (index-1) limit
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// partial sieve loop, adjusting larges/smalls, compressing larges/roughs...</span>
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<span style="color: #008080;">while</span> <span style="color: #000000;">sqri</span><span style="color: #0000FF;"><=</span><span style="color: #000000;">mxndx</span> <span style="color: #008080;">do</span> <span style="color: #000080;font-style:italic;">// partial sieve to square index limit</span>
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<span style="color: #008080;">if</span> <span style="color: #008080;">not</span> <span style="color: #000000;">composite</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #008080;">then</span>
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<span style="color: #000080;font-style:italic;">// cull from composite so they will never be found again</span>
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<span style="color: #000000;">composite</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">true</span> <span style="color: #000080;font-style:italic;">// cull bp and multiples</span>
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<span style="color: #008080;">for</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">=</span><span style="color: #000000;">sqri</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">mxndx</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">by</span> <span style="color: #000000;">bp</span> <span style="color: #008080;">do</span>
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<span style="color: #000000;">composite</span><span style="color: #0000FF;">[</span><span style="color: #000000;">c</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">true</span>
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
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<span style="color: #000080;font-style:italic;">// partial sieving to current base prime is now completed!
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// now adjust `larges` for latest partial sieve pass...</span>
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<span style="color: #004080;">integer</span> <span style="color: #000000;">ori</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span> <span style="color: #000080;font-style:italic;">// compress input rough index(k) to output one</span>
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<span style="color: #008080;">for</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">,</span><span style="color: #000000;">q</span> <span style="color: #008080;">in</span> <span style="color: #000000;">roughs</span> <span style="color: #008080;">to</span> <span style="color: #000000;">mxri</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
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<span style="color: #000080;font-style:italic;">// q is not necessarily prime but may be a product of primes not yet
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// culled by partial sieving (saves ops cmprd to recursive Legendre)
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// skip over values of `q` already culled in the last partial sieve:</span>
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<span style="color: #004080;">integer</span> <span style="color: #000000;">qi</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">q</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">;</span> <span style="color: #000080;font-style:italic;">// index of always odd q!</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #008080;">not</span> <span style="color: #000000;">composite</span><span style="color: #0000FF;">[</span><span style="color: #000000;">qi</span><span style="color: #0000FF;">]</span> <span style="color: #008080;">then</span>
|
||||
<span style="color: #000080;font-style:italic;">// since `q` cannot be equal to bp due to cull of bp and above skip;</span>
|
||||
<span style="color: #004080;">atom</span> <span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">bp</span><span style="color: #0000FF;">*</span><span style="color: #000000;">q</span><span style="color: #0000FF;">,</span> <span style="color: #000080;font-style:italic;">// `d` is odd product of some combination of odd primes!
|
||||
// the following computation is essential to the algorithm's speed,
|
||||
// see the Nim entry for the full details of how this works</span>
|
||||
<span style="color: #000000;">dadj</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;"><=</span><span style="color: #000000;">sqrtn</span> <span style="color: #0000FF;">?</span> <span style="color: #000000;">larges</span><span style="color: #0000FF;">[</span><span style="color: #000000;">smalls</span><span style="color: #0000FF;">[</span><span style="color: #000000;">half</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">)]-</span><span style="color: #000000;">nbp</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
|
||||
<span style="color: #0000FF;">:</span> <span style="color: #000000;">smalls</span><span style="color: #0000FF;">[</span><span style="color: #000000;">half</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">/</span><span style="color: #000000;">d</span><span style="color: #0000FF;">))])</span>
|
||||
<span style="color: #000000;">ori</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
|
||||
<span style="color: #000000;">larges</span><span style="color: #0000FF;">[</span><span style="color: #000000;">ori</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">larges</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]-</span><span style="color: #000000;">dadj</span><span style="color: #0000FF;">+</span><span style="color: #000000;">nbp</span> <span style="color: #000080;font-style:italic;">// base primes count over subtracted!
|
||||
// eliminate rough values that have been culled in partial sieve:
|
||||
// note that `larges` and `roughs` indices relate to each other!</span>
|
||||
<span style="color: #000000;">roughs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">ori</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">q</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
|
||||
|
||||
<span style="color: #004080;">integer</span> <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">mxndx</span> <span style="color: #000080;font-style:italic;">// and adjust `smalls` for latest partial sieve pass...
|
||||
// this is faster than recounting over the `composite` array for each loop...</span>
|
||||
<span style="color: #008080;">for</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">=(</span><span style="color: #000000;">sqrtn</span><span style="color: #0000FF;">/</span><span style="color: #000000;">bp</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)||</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">bp</span> <span style="color: #008080;">by</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">2</span> <span style="color: #008080;">do</span> <span style="color: #000080;font-style:italic;">// k always odd!
|
||||
// `c` is correction from current count to desired count...
|
||||
// `e` is end limit index no correction is necessary for current cull...</span>
|
||||
<span style="color: #004080;">integer</span> <span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">smalls</span><span style="color: #0000FF;">[</span><span style="color: #000000;">half</span><span style="color: #0000FF;">(</span><span style="color: #000000;">k</span><span style="color: #0000FF;">)]-</span><span style="color: #000000;">nbp</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #000000;">e</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">((</span><span style="color: #000000;">k</span><span style="color: #0000FF;">*</span><span style="color: #000000;">bp</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">while</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">>=</span><span style="color: #000000;">e</span> <span style="color: #008080;">do</span>
|
||||
<span style="color: #000080;font-style:italic;">-- smalls[m+1] -= c -- (grr, js! [I have a plan, working on it])</span>
|
||||
<span style="color: #000000;">m</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
|
||||
<span style="color: #000000;">smalls</span><span style="color: #0000FF;">[</span><span style="color: #000000;">m</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">c</span>
|
||||
<span style="color: #000080;font-style:italic;">-- m -= 1</span>
|
||||
<span style="color: #000000;">m</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">2</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
|
||||
|
||||
<span style="color: #000000;">nbp</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span> <span style="color: #000080;font-style:italic;">// increase number of found base primes</span>
|
||||
<span style="color: #000000;">mxri</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">ori</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #000080;font-style:italic;">// advance rough index for later</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<span style="color: #000000;">bp</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">2</span>
|
||||
<span style="color: #000000;">sqri</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)*(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">i</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
|
||||
|
||||
<span style="color: #000080;font-style:italic;">// now `smalls` is a LUT of odd prime accumulated counts for all odd primes;
|
||||
// `roughs` is exactly the "k-roughs" up to the sqrt of range with `k` (erm,
|
||||
// mxri?) the index of the next prime above the quad root of the range;
|
||||
// `larges` is the partial prime counts for each of the `roughs` values...
|
||||
// note that `larges` values include the count of the odd base primes!!!
|
||||
// - and `composite` is never used again!
|
||||
|
||||
// the following does the top-most "phi tree" calculation:
|
||||
// the answer to here is all valid `phis`, combined here by subtraction,
|
||||
// + compensate for included odd base prime counts over subracted above:</span>
|
||||
<span style="color: #004080;">atom</span> <span style="color: #000000;">result</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">larges</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">-</span> <span style="color: #7060A8;">sum</span><span style="color: #0000FF;">(</span><span style="color: #000000;">larges</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">..</span><span style="color: #000000;">mxri</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">])</span>
|
||||
<span style="color: #0000FF;">+</span> <span style="color: #7060A8;">trunc</span><span style="color: #0000FF;">((</span><span style="color: #000000;">mxri</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*(</span><span style="color: #000000;">nbp</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">))*</span><span style="color: #000000;">mxri</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #0000FF;">+</span> <span style="color: #000000;">1</span> <span style="color: #000080;font-style:italic;">// include the only even prime, ie 2
|
||||
|
||||
// This loop adds the counts due to the products of the `roughs` primes,
|
||||
// of which we only use two different ones at a time, as all the
|
||||
// combinations with lower primes than the cube root of the range have
|
||||
// already been computed and included with the previous major loop...
|
||||
// see text description in the Nim entry for how this works...</span>
|
||||
<span style="color: #008080;">for</span> <span style="color: #000000;">ri</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span> <span style="color: #008080;">in</span> <span style="color: #000000;">roughs</span> <span style="color: #008080;">from</span> <span style="color: #000000;">2</span> <span style="color: #008080;">do</span> <span style="color: #000080;font-style:italic;">// for all `roughs` (now prime) bar '1':</span>
|
||||
<span style="color: #004080;">atom</span> <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">trunc</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">/</span><span style="color: #000000;">p</span><span style="color: #0000FF;">),</span> <span style="color: #000080;font-style:italic;">// `m` is the `p` quotient
|
||||
// so that the end limit `e` can be calculated based on `n`/(`p`^2)</span>
|
||||
<span style="color: #000000;">e</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">smalls</span><span style="color: #0000FF;">[</span><span style="color: #000000;">half</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">/</span><span style="color: #000000;">p</span><span style="color: #0000FF;">))+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]-</span><span style="color: #000000;">nbp</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span>
|
||||
<span style="color: #000080;font-style:italic;">// the following test is equivalent to non-splitting optmization:</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #000000;">e</span><span style="color: #0000FF;"><=</span><span style="color: #000000;">ri</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #000080;font-style:italic;">// quit when no more pairs! - aka stop
|
||||
// at about `p` of cube root of range!</span>
|
||||
<span style="color: #008080;">for</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">=</span><span style="color: #000000;">ri</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">e</span> <span style="color: #008080;">do</span> <span style="color: #000080;font-style:italic;">// for all `roughs` greater than `p` to limit:</span>
|
||||
<span style="color: #000000;">result</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">smalls</span><span style="color: #0000FF;">[</span><span style="color: #000000;">half</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">/</span><span style="color: #000000;">roughs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]))];</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
|
||||
<span style="color: #000080;font-style:italic;">// compensate for all the extra base prime counts just added!</span>
|
||||
<span style="color: #000000;">result</span> <span style="color: #0000FF;">-=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">e</span><span style="color: #0000FF;">-</span><span style="color: #000000;">ri</span><span style="color: #0000FF;">)*(</span><span style="color: #000000;">nbp</span><span style="color: #0000FF;">+</span><span style="color: #000000;">ri</span><span style="color: #0000FF;">-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
|
||||
<span style="color: #008080;">return</span> <span style="color: #000000;">result</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
|
||||
|
||||
<span style="color: #004080;">atom</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()</span>
|
||||
<span style="color: #008080;">constant</span> <span style="color: #000000;">expected</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">4</span><span style="color: #0000FF;">,</span><span style="color: #000000;">25</span><span style="color: #0000FF;">,</span><span style="color: #000000;">168</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1229</span><span style="color: #0000FF;">,</span><span style="color: #000000;">9592</span><span style="color: #0000FF;">,</span><span style="color: #000000;">78498</span><span style="color: #0000FF;">,</span><span style="color: #000000;">664579</span><span style="color: #0000FF;">,</span><span style="color: #000000;">5761455</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #000000;">50847534</span><span style="color: #0000FF;">,</span><span style="color: #000000;">455052511</span><span style="color: #0000FF;">,</span><span style="color: #000000;">4118054813</span><span style="color: #0000FF;">,</span><span style="color: #000000;">37607912018</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #000000;">346065536839</span><span style="color: #0000FF;">,</span><span style="color: #000000;">3204941750802</span><span style="color: #0000FF;">}</span>
|
||||
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">iff</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">platform</span><span style="color: #0000FF;">()=</span><span style="color: #004600;">JS</span><span style="color: #0000FF;">?</span><span style="color: #000000;">11</span><span style="color: #0000FF;">:</span><span style="color: #000000;">14</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span> <span style="color: #000080;font-style:italic;">-- (sp: keep js under 2s)</span>
|
||||
<span style="color: #004080;">atom</span> <span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">count_primes</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">10</span><span style="color: #0000FF;">,</span><span style="color: #000000;">i</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #7060A8;">assert</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">==</span><span style="color: #000000;">expected</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">])</span>
|
||||
<span style="color: #004080;">string</span> <span style="color: #000000;">e</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">elapsed</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">time</span><span style="color: #0000FF;">()-</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">0.1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" (%s)"</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"10^%d = %d%s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">i</span><span style="color: #0000FF;">,</span><span style="color: #000000;">c</span><span style="color: #0000FF;">,</span><span style="color: #000000;">e</span><span style="color: #0000FF;">})</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
|
||||
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\nTook %s\n"</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">elapsed</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">time</span><span style="color: #0000FF;">()-</span><span style="color: #000000;">t</span><span style="color: #0000FF;">))</span>
|
||||
<!--
|
||||
Loading…
Add table
Add a link
Reference in a new issue