crank conjecture

The fraction of odd values of the partition function P(n) is roughly 50%, independent of n, whereas odd values of Q(n) occur with ever decreasing frequency as n becomes large. Kolberg proved that there are infinitely many even and odd values of P(n). Leibniz noted that P(n) is prime for n = 2, 3, 4, 5, 6, but not 7. In fact, values of n for which P(n) is prime are 2, 3, 4, 5, 6, 13, 36, 77, 132, 157, 168, 186, ... (OEIS A046063), corresponding to 2, 3, 5, 7, 11, 101, 17977, 10619863, ... (OEIS A049575). Numbers which cannot be written as a product of P(n) are 13, 17, 19, 23, 26, 29, 31, 34, 37, 38, 39, ... (OEIS A046064), corresponding to numbers of nonisomorphic Abelian groups which are not possible for any group order.

congruence | Erdős-Ivić conjecture | integer sequence primes | Newman's conjecture | partition function P | partition function q | partition function Q | partition function Q congruences