CONGRUENCE IDENTITIES INVOLVING SUMS OF ODD DIVISORS FUNCTION

In this paper, inspired by a classical connections between partitions and divisors, we investigate some congruence identities involving sums of the odd divisor function sigma(odd) (n) which is defined by sigma(odd) (n) = Sigma(d\n)(d odd) d. In this context, we conjectured that the congruence Sigma(infinity)(k=-infinity) sigma(odd) (n-k(3k - 1)/2 equivalent to{n (mod m); if n = j (3 j - 1)/2, j is an element of Z, 0 (mod m), otherwise. is valid for any positive integer n if and only if m is an element of {2,3,6}