On a conjecture of Nicolas-Sarkozy about partitions

[摘要] Let N be the set of positive integers, B = {b(1) < .. N, where p(A, n) denotes the number of partitions of n with parts in A. Let us denote by sigma(A, n) the sum of the divisors of n belonging to A. In this paper, we prove that sigma(A, 2n) mod 4 is periodic with period q(2) multiple of q period of sigma(A, n) mod 2; we also give the sets B subset of {1,...,5} and the values of N, N less than or equal to 10, for which q(2) not equal q. Moreover, we show that if A(x) is the counting function of A then for A = A(0)({1,2,3},3), lim(x-->infinity)A(x)/x less than or equal to 1/4. (C) 2002 Elsevier Science (USA).

[发布日期] 2002-08-01

[关键词] partitions;congruence;period;primes