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j(^r) = E c(n)xn = (1 +240 0 u3(n)xn) /xf24(x) 744, where 03(n) = d3. dln Then p(n) is just the number of unrestricted partitions of n, and c(n) is the Fourier coefficient of Klein's modular invariant. For primes q 11 this type(1) of congruence does not persist. Newman [6], in the context of p(n), has proposed the question: (3) Given a, m, is p(n) -a (mod m) soluble for an infinity of n ? We may ask further: (4) Given a, m, is p(n) _ a (mod m) soluble for values of n with positive density ? The best hope of establishing (3) seems to be that one may exhibit explicit congruences of the form (5) p(h(n)) _ a (mod m),

Some properties of $p(n)$ and $c(n)$ modulo powers of $13$