<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\overline{p}(n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi>p</mi> <mo>¯</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the number of overpartitions of <i>n</i>. In this note, we find some congruences for this partition function that appear to have been overlooked in the existing literature. For example, we prove that for all <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n \ge 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation><Equation ID="Equ39"> <EquationSource Format="TEX">\(\begin{aligned} \overline{p}\left( 648n+594\right) \equiv 0 \pmod {2^7}. \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mover> <mi>p</mi> <mo>¯</mo> </mover> <mfenced close=")" open="("> <mn>648</mn> <mi>n</mi> <mo>+</mo> <mn>594</mn> </mfenced> <mo>≡</mo> <mn>0</mn> <mspace width="10.0pt" /> <mrow> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <msup> <mn>2</mn> <mn>7</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>We also propose a conjectural congruence modulo <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(5^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>5</mn> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>, which is one in a handful of its type. Our approach is elementary dissection techniques.</p>

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Missed congruences of overpartitions

  • Hirakjyoti Das,
  • Hemjyoti Nath

摘要

Let \(\overline{p}(n)\) p ¯ ( n ) denote the number of overpartitions of n. In this note, we find some congruences for this partition function that appear to have been overlooked in the existing literature. For example, we prove that for all \(n \ge 0\) n 0 \(\begin{aligned} \overline{p}\left( 648n+594\right) \equiv 0 \pmod {2^7}. \end{aligned}\) p ¯ 648 n + 594 0 ( mod 2 7 ) . We also propose a conjectural congruence modulo \(5^2\) 5 2 , which is one in a handful of its type. Our approach is elementary dissection techniques.