Data update
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Ingy döt Net
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2347 changed files with 62432 additions and 16731 deletions
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@ -8,19 +8,15 @@ for n = 2 to 1000
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.
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print "First 99 Mertens numbers:"
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write " "
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numfmt 0 2
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numfmt 2 0
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for n = 1 to 99
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write mertens[n] & " "
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if n mod 10 = 9
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print ""
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.
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if n mod 10 = 9 : print ""
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.
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for n = 1 to 1000
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if mertens[n] = 0
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zeros += 1
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if mertens[n - 1] <> 0
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crosses += 1
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.
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if mertens[n - 1] <> 0 : crosses += 1
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.
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.
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print ""
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@ -1,61 +1,118 @@
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/*REXX pgm computes & shows a value grid of the Mertens function for a range of integers*/
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parse arg LO HI grp eqZ xZ . /*obtain optional arguments from the CL*/
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if LO=='' | LO=="," then LO= 0 /*Not specified? Then use the default.*/
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if HI=='' | HI=="," then HI= 199 /* " " " " " " */
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if grp=='' | grp=="," then grp= 20 /* " " " " " " */
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if eqZ=='' | eqZ=="," then eqZ= 1000 /* " " " " " " */
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if xZ=='' | xZ=="," then xZ= 1000 /* " " " " " " */
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call genP /*generate primes up to max √ HIHI */
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call Franz LO, HI
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if eqZ>0 then call Franz 1, -eqZ
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if xZ>0 then call Franz -1, xZ
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exit 0 /*stick a fork in it, we're all done. */
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/*──────────────────────────────────────────────────────────────────────────────────────*/
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Franz: parse arg a 1 oa,b 1 ob; @Mertens= ' The Mertens sequence from '
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a= abs(a); b= abs(b); grid= oa>=0 & ob>=0 /*semaphore used to show a grid title. */
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if grid then say center(@Mertens LO " ──► " HI" ", max(50, grp*3), '═') /*show title*/
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else say
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zeros= 0 /*# of 0's found for Mertens function.*/
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Xzero= 0 /*number of times that zero was crossed*/
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$=; prev= /*$ holds output grid of GRP numbers. */
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do j=a to b; _= Mertens(j) /*process some numbers from LO ──► HI.*/
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if _==0 then zeros= zeros + 1 /*Is Zero? Then bump the zeros counter*/
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if _==0 then if prev\==0 then Xzero= Xzero+1 /*prev ¬=0? " " " Xzero " */
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prev= _
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if grid then $= $ right(_, 2) /*build grid if A & B are non─negative.*/
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if words($)==grp then do; say substr($, 2); $= /*show grid if fully populated, */
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end /* and nullify it for more #s. */
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end /*j*/ /*for small grids, using wordCnt is OK.*/
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-- 25 Apr 2025
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include Settings
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arg xx
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if xx = '' then
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xx = 99
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if $\=='' then say substr($, 2) /*handle any residual numbers not shown*/
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if oa<0 then say @Mertens a " to " b ' has crossed zero ' Xzero " times."
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if ob<0 then say @Mertens a " to " b ' has ' zeros " zeros."
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say 'MERTENS FUNCTION'
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say version
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say
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call Tasks1 xx
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call Tasks2 xx
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call Tasks3 xx
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exit
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Tasks1:
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procedure expose squa.
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arg xx
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call Time('r')
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say 'Tasks using Sieve...'
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call Squarefrees 2*xx
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if xx < 100 then
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say 'M1 thru M'xx
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e = 0; o = 0; a = 0; b = 0
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n = 1; s = squa.1; p = s
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do i = 1 to xx
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if s <= i then do
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if s = 1 then
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f = 0
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else
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f = Factors(s)
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e = e+Even(f); o = o+Odd(f); m = e-o
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n = n+1; s = squa.n
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end
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if xx < 100 & i < 100 then do
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call Charout ,Right(m,3)
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if i//10 = 0 then
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say
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end
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if m = 0 then do
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a = a+1
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if p <> 0 then
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b = b + 1
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end
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p = m
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end
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say
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say 'M1 thru M'xx 'equals zero' a 'times'
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say 'M1 thru M'xx 'crosses zero' b 'times'
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say Format(Time('e'),,3) 'seconds'; say
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return
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/*──────────────────────────────────────────────────────────────────────────────────────*/
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Mertens: procedure expose @. !!. M.; parse arg n; if M.n\==. then return M.n
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if n<1 then return '∙'; m= 0 /*handle special cases of non─positive#*/
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do k=1 for n; m= m + mobius(k) /*sum the MU's up to N. */
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end /*k*/ /* [↑] mobius function uses memoization*/
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M.n= m; return m /*return the sum of all the MU's. */
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/*──────────────────────────────────────────────────────────────────────────────────────*/
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mobius: procedure expose @. !!.; parse arg x 1 ox /*get integer to be tested for mu */
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if !!.x\==. then return !!.x /*X computed before? Return that value*/
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if x<1 then return '∙'; mu= 0 /*handle special case of non-positive #*/
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do k=1; p= @.k; if p>x then leave /* (P) > X? Then we're done.*/
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if p*p>x then do; mu= mu+1; leave; end /* (P**2) > X? Bump # and leave*/
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if x//p==0 then do; mu= mu+1 /*X divisible by P? Bump mu number.*/
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x= x % p /* Divide by prime. */
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if x//p==0 then return 0 /*X÷by P? Then return zero*/
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end
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end /*k*/ /*MU (below) is almost always small, <9*/
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!!.ox= -1 ** mu; return !!.ox /*raise -1 to the mu power, memoize it.*/
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/*──────────────────────────────────────────────────────────────────────────────────────*/
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genP: @.1=2; @.2=3; @.3=5; @.4=7; @.5=11; @.6=13 /*initialize some low primes; # primes.*/
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!!.=.; M.=!!.; #= 6; sq.#= @.6**2 /* " 2 arrays for memoization. */
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do j=@.#+4 by 2 to max(HI, eqZ, xZ); parse var j '' -1 _ /*odd Ps from now on*/
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if _==5 then iterate; if j//3==0 then iterate; if j//7==0 then iterate /*÷ 5 3 7*/
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do k=7 while sq.k<=j /*divide by some generated odd primes. */
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if j//@.k==0 then iterate j /*Is J divisible by P? Then not prime*/
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end /*k*/ /* [↓] a prime (J) has been found. */
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#= #+1; @.#=j; sq.j= j*j /*bump P count; P──►@.; compute J**2*/
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end /*j*/; return /*calculate the squares of some primes.*/
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Tasks2:
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procedure expose squa.
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arg xx
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call Time('r')
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say 'Tasks using Moebius function...'
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if xx < 100 then
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say 'M1 thru M'xx
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e = 0; o = 0; a = 0; b = 0
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s = 0
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do i = 1 to xx
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s = s+Moebius(i)
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if xx < 100 & i < 100 then do
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call Charout ,Right(s,3)
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if i//10 = 0 then
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say
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end
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if s = 0 then do
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a = a+1
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if p <> 0 then
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b = b + 1
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end
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p = s
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end
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say
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say 'M1 thru M'xx 'equals zero' a 'times'
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say 'M1 thru M'xx 'crosses zero' b 'times'
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say Format(Time('e'),,3) 'seconds'; say
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return
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Tasks3:
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procedure expose squa.
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arg xx
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call Time('r')
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say 'Tasks using Squarefree function...'
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if xx < 100 then
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say 'M1 thru M'xx
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e = 0; o = 0; a = 0; b = 0
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do s = 1 to xx
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if Squarefree(s) then do
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if s = 1 then
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f = 0
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else
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f = Factors(s)
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e = e+Even(f); o = o+Odd(f); m = e-o
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end
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if xx < 100 & s < 100 then do
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call Charout ,Right(m,3)
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if s//10 = 0 then
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say
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end
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if m = 0 then do
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a = a+1
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if p <> 0 then
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b = b + 1
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end
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p = m
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end
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say
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say 'M1 thru M'xx 'equals zero' a 'times'
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say 'M1 thru M'xx 'crosses zero' b 'times'
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say Format(Time('e'),,3) 'seconds'; say
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return
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include Numbers
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include Sequences
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include Functions
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include Abend
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