On the divisor function of sets with even partition functions |
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Authors: | N Baccar F Ben Sa?d A Zekraoui |
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Institution: | (1) Université de Monastir, Faculté des Sciences de Monastir, Département de Mathématiques, Avenue de l'environnement, 5000, Monastir, Tunisie;(2) Université de Monastir, Faculté des Sciences de Monastir, Département de Mathématiques, Avenue de l'environnement, 5000, Monastir, Tunisie;(3) Université de Monastir, Faculté des Sciences de Monastir, Département de Mathématiques, Avenue de l'environnement, 5000, Monastir, Tunisie |
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Abstract: | Summary For P∈ F2z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of N (cf. 9]) such that Σn≧0 p(A,n)zn ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ d|n, d∈A d, namely, the periodicity of the sequences (σ(A,2kn) mod 2k+1)n≧1 for all k ≧ 0 which was proved in 3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula
to σ(A,2kn) mod 2k+1 will be established, from which it will be shown that the weight σ(A1,2kzi) mod 2k+1 on the orbit <InlineEquation ID=IE"1"><EquationSource Format="TEX"><!CDATA<InlineEquation ID=IE"2"><EquationSource Format="TEX"><!CDATA$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>z_i$
is moved on some other orbit zj when A1 is replaced by A2 with A1= A(P1) and A2= A(P2) P1 and P2 being irreducible in F2z] of the same odd order. |
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Keywords: | cyclotomic polynomials partitions order of a polynomial orbits symmetric functions periodic sequences the Graeffe transformation |
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