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<p>The partition function <i>SOME</i>(<i>n</i>) is defined as the sum of all the odd parts in the partitions of <i>n</i> minus the sum of all the even parts in the partitions of <i>n</i>. The generating function for <i>SOME</i>(<i>n</i>) is derived, and congruences for <i>SOME</i>(<i>n</i>) are found; for instance, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(5 \mid SOME(5n+4)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>5</mn> <mo>∣</mo> <mi>S</mi> <mi>O</mi> <mi>M</mi> <mi>E</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mi>n</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. This implies that the sum of all the odd parts in the partitions of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(5n+4\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>5</mn> <mi>n</mi> <mo>+</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> is divisible by 5, and the same holds for the sum of the even parts.</p>

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\(p(5n+4)\) again

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月3日

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  • George E. Andrews,
  • Manosij Ghosh Dastidar

摘要

The partition function SOME(n) is defined as the sum of all the odd parts in the partitions of n minus the sum of all the even parts in the partitions of n. The generating function for SOME(n) is derived, and congruences for SOME(n) are found; for instance, \(5 \mid SOME(5n+4)\) 5 ∣ S O M E ( 5 n + 4 ) . This implies that the sum of all the odd parts in the partitions of \(5n+4\) 5 n + 4 is divisible by 5, and the same holds for the sum of the even parts.