64 lines
2.8 KiB
Text
64 lines
2.8 KiB
Text
The [[wp:Mian–Chowla sequence|Mian–Chowla sequence]] is an integer sequence defined recursively.
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Mian–Chowla is an infinite instance of a [[wp:Sidon sequence|Sidon sequence]], and belongs to the class known as B₂ sequences.
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The sequence starts with:
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::<span style="font-style:italic;font-weight:bold;font-size:125%;">a<sub>1</sub> = 1</span>
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then for <span style="font-style:italic;font-weight:bold;font-size:125%;padding:.3em;">n > 1,</span> <span style="font-style:italic;font-weight:bold;font-size:125%;padding:.3em;">a<sub>n</sub></span> is the smallest positive integer such that every pairwise sum
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::<span style="font-style:italic;font-weight:bold;font-size:125%;">a<sub>i</sub> + a<sub>j</sub>
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is distinct, for all <span style="font-style:italic;font-weight:bold;font-size:125%;">i</span> and <span style="font-style:italic;font-weight:bold;font-size:125%;">j</span> less than or equal to <span style="font-style:italic;font-weight:bold;font-size:125%;">n</span>.
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;The Task:
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:* Find and display, here, on this page the first 30 terms of the Mian–Chowla sequence.
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:* Find and display, here, on this page the 91st through 100th terms of the Mian–Chowla sequence.
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Demonstrating working through the first few terms longhand:
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::<span style="font-style:italic;font-weight:bold;font-size:125%;">a<sub>1</sub> = 1</span>
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::<span style="font-weight:bold;">1 + 1 = 2</span>
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Speculatively try <span style="font-style:italic;font-weight:bold;font-size:125%;padding:.3em;">a<sub>2</sub> = 2</span>
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::<span style="font-weight:bold;">1 + 1 = 2</span>
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::<span style="font-weight:bold;">1 + 2 = 3</span>
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::<span style="font-weight:bold;">2 + 2 = 4</span>
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There are no repeated sums so '''2''' is the next number in the sequence.
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Speculatively try <span style="font-style:italic;font-weight:bold;font-size:125%;padding:.3em;">a<sub>3</sub> = 3</span>
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::<span style="font-weight:bold;">1 + 1 = 2</span>
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::<span style="font-weight:bold;">1 + 2 = 3</span>
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::<span style="font-weight:bold;">1 + 3 = <span style="background-color:yellow;">4</span></span>
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::<span style="font-weight:bold;">2 + 2 = <span style="background-color:yellow;">4</span></span>
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::<span style="font-weight:bold;">2 + 3 = 5</span>
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::<span style="font-weight:bold;">3 + 3 = 6</span>
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Sum of '''4''' is repeated so '''3''' is rejected.
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Speculatively try <span style="font-style:italic;font-weight:bold;font-size:125%;padding:.3em;">a<sub>3</sub> = 4</span>
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::<span style="font-weight:bold;">1 + 1 = 2</span>
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::<span style="font-weight:bold;">1 + 2 = 3</span>
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::<span style="font-weight:bold;">1 + 4 = 5</span>
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::<span style="font-weight:bold;">2 + 2 = 4</span>
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::<span style="font-weight:bold;">2 + 4 = 6</span>
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::<span style="font-weight:bold;">4 + 4 = 8</span>
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There are no repeated sums so '''4''' is the next number in the sequence.
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And so on...
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;See also:
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:* [[oeis:A005282|OEIS:A005282 Mian-Chowla sequence]]
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