<p>Recently, two functions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\sigma _{o}\textrm{mex}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mi>o</mi> </msub> <mtext>mex</mtext> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\sigma _{e}\textrm{mex}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mi>e</mi> </msub> <mtext>mex</mtext> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> which denote the sum of odd minimal excludants and the sum of even minimal excludants, respectively, were introduced by Baruah, Bhoria, Eyyunni and Maji. They also proved some congruences on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\sigma _{o}\textrm{mex}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mi>o</mi> </msub> <mtext>mex</mtext> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\sigma _{e}\textrm{mex}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mi>e</mi> </msub> <mtext>mex</mtext> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> modulo 4 and 8 and posed two conjectural congruences on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\sigma _{o}\textrm{mex}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mi>o</mi> </msub> <mtext>mex</mtext> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\sigma _{e}\textrm{mex}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mi>e</mi> </msub> <mtext>mex</mtext> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> which were first confirmed by Du and Tang based on the theory of modular forms. Very recently, Singh and Barman proved infinite families of congruences modulo 4 and 8 for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\sigma _{o}\textrm{mex}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mi>o</mi> </msub> <mtext>mex</mtext> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\sigma _{e}\textrm{mex}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mi>e</mi> </msub> <mtext>mex</mtext> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Motivated by their work, we establish infinite families of congruences modulo 8 and 16 for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\sigma _{o}\textrm{mex}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mi>o</mi> </msub> <mtext>mex</mtext> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\sigma _{e}\textrm{mex}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mi>e</mi> </msub> <mtext>mex</mtext> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in this paper. In particular, the two conjectural congruences are corollaries of our results.</p>

New congruences on the sum of odd and even minimal excludants

期刊论文
发表日期：

2026年3月21日

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Si Jia Wang,
Eric H. Liu,
Olivia X. M. Yao

摘要

Recently, two functions $\sigma _{o}\textrm{mex}(n)$ $σ_{o} mex (n)$ and $\sigma _{e}\textrm{mex}(n)$ $σ_{e} mex (n)$ which denote the sum of odd minimal excludants and the sum of even minimal excludants, respectively, were introduced by Baruah, Bhoria, Eyyunni and Maji. They also proved some congruences on $\sigma _{o}\textrm{mex}(n)$ $σ_{o} mex (n)$ and $\sigma _{e}\textrm{mex}(n)$ $σ_{e} mex (n)$ modulo 4 and 8 and posed two conjectural congruences on $\sigma _{o}\textrm{mex}(n)$ $σ_{o} mex (n)$ and $\sigma _{e}\textrm{mex}(n)$ $σ_{e} mex (n)$ which were first confirmed by Du and Tang based on the theory of modular forms. Very recently, Singh and Barman proved infinite families of congruences modulo 4 and 8 for $\sigma _{o}\textrm{mex}(n)$ $σ_{o} mex (n)$ and $\sigma _{e}\textrm{mex}(n)$ $σ_{e} mex (n)$ . Motivated by their work, we establish infinite families of congruences modulo 8 and 16 for $\sigma _{o}\textrm{mex}(n)$ $σ_{o} mex (n)$ and $\sigma _{e}\textrm{mex}(n)$ $σ_{e} mex (n)$ in this paper. In particular, the two conjectural congruences are corollaries of our results.