login
A328170
Number of integer partitions of n whose parts minus 1 are relatively prime.
12
0, 0, 1, 1, 2, 3, 5, 8, 12, 18, 27, 38, 53, 74, 102, 137, 184, 241, 317, 413, 536, 687, 880, 1112, 1405, 1765, 2215, 2755, 3424, 4229, 5216, 6402, 7847, 9572, 11662, 14148, 17139, 20688, 24940, 29971, 35969, 43044, 51438, 61311, 72985, 86678, 102807, 121675
OFFSET
0,5
COMMENTS
A partition is relatively prime if the GCD of its parts is 1. Zeros are ignored when computing GCD, and the empty set has GCD 0.
LINKS
FORMULA
G.f.: Sum_{d>=1} mu(d)*(-1/(1-x) + 1/(Prod_{k>=0} 1 - x^(k*d + 1))). - Andrew Howroyd, Oct 17 2019
EXAMPLE
The a(2) = 1 through a(9) = 18 partitions:
(2) (21) (22) (32) (42) (43) (62) (54)
(211) (221) (222) (52) (332) (63)
(2111) (321) (322) (422) (72)
(2211) (421) (431) (432)
(21111) (2221) (521) (522)
(3211) (2222) (621)
(22111) (3221) (3222)
(211111) (4211) (3321)
(22211) (4221)
(32111) (4311)
(221111) (5211)
(2111111) (22221)
(32211)
(42111)
(222111)
(321111)
(2211111)
(21111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], GCD@@(#-1)==1&]], {n, 0, 30}]
PROG
(PARI) seq(n)=Vec(sum(d=1, n-1, moebius(d)*(-1/(1-x) + 1/prod(k=0, n\d, 1 - x*x^(k*d) + O(x*x^n)))), -(n+1)) \\ Andrew Howroyd, Oct 17 2019
CROSSREFS
The Heinz numbers of these partitions are given by A328168.
Partitions whose parts are relatively prime are A000837.
Partitions whose parts plus 1 are relatively prime are A318980.
The GCD of the prime indices of n, all minus 1, is A328167(n).
Sequence in context: A280278 A136275 A344002 * A078408 A007478 A014605
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 09 2019
STATUS
approved