A318980 Number of integer partitions of n whose parts plus 1 are relatively prime.

0, 0, 1, 1, 4, 5, 9, 13, 21, 29, 43, 56, 79, 109, 146, 192, 254, 329, 428, 553, 707, 900, 1139, 1434, 1800, 2251, 2799, 3472, 4286, 5275, 6469, 7918, 9655, 11755, 14252, 17248, 20817, 25084, 30134, 36142, 43235, 51644, 61548, 73241, 86961, 103108, 122010

Offset: 1

Author

Gus Wiseman, Sep 06 2018

Keywords

• nonn

Examples

			The a(7) = 9 partitions are (61), (43), (421), (4111), (322), (3211), (2221), (22111), (211111).
The a(8) = 13 partitions:
  (62),
  (332), (422), (431), (521), (611),
  (3221), (4211),
  (22211), (32111), (41111),
  (221111),
  (2111111).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000837, A007359, A018783, A051424, A281116, A289508, A289509, A318978.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],GCD@@(#+1)==1&]],{n,30}]

PARI

seq(n)={Vec(sum(d=1, n+1, moebius(d)*(-1 + 1/prod(k=ceil(2/d), (n+1)\d, 1 - x^(k*d-1) + O(x*x^n)))), -n)} \\ Andrew Howroyd, Oct 17 2019

Formula

G.f.: Sum_{d>=1} mu(d)*(-1 + 1/(Prod_{k>=2/d} 1 - x^(k*d - 1))). - Andrew Howroyd, Oct 17 2019