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

1,5

Links

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

Formula

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

Example

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).

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A116045 A096818 A282467 * A226622 A332302 A287209

Adjacent sequences: A318977 A318978 A318979 * A318981 A318982 A318983

Keyword

nonn

Author

Gus Wiseman, Sep 06 2018

Status

approved