A161077 Number of partitions of n into primes or 1 where every part appears at least 2 times.

0, 1, 1, 2, 1, 4, 2, 5, 5, 8, 6, 12, 10, 16, 16, 22, 21, 31, 30, 40, 42, 53, 55, 71, 73, 90, 96, 116, 123, 149, 157, 186, 200, 234, 250, 292, 312, 360, 388, 443, 476, 543, 584, 660, 712, 800, 862, 967, 1041, 1160, 1252, 1390, 1497, 1659, 1786, 1970, 2124, 2336, 2513, 2760

Offset

1,4

Links

R. H. Hardin, Table of n, a(n) for n = 1..1000

Formula

G.f.: -1+(1+x^2/(1-x))*product[1+x^{2*p(j)}/(1-x^{p(j)}, j=1..infinity], where p(j) is the j-th prime. - Emeric Deutsch, Jun 27 2009

Example

a(8)=5 because we have 3311, 2222, 22211, 221111, and 1^8. - Emeric Deutsch, Jun 27 2009

Maple

g := -1+(1+x^2/(1-x))*(product(1+x^(2*ithprime(j))/(1-x^ithprime(j)), j = 1 .. 20)): gser := series(g, x = 0, 70): seq(coeff(gser, x, n), n = 1 .. 60); # Emeric Deutsch, Jun 27 2009

Mathematica

nmax = 100; Rest[CoefficientList[Series[-1 + (1 + x^2/(1-x)) * Product[1 + x^(2*Prime[k]) / (1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Nov 28 2020 *)

Crossrefs

Sequence in context: A161307 A161241 A161026 * A339220 A161293 A328163

Adjacent sequences: A161074 A161075 A161076 * A161078 A161079 A161080

Keyword

nonn

Author

R. H. Hardin, Jun 02 2009

Extensions

Definition edited to "primes or 1" by R. H. Hardin, Jun 22 2009

Status

approved