A322968 Number of integer partitions of n with no ones whose parts are all powers of the same squarefree number.

1, 0, 1, 1, 2, 1, 4, 1, 4, 2, 6, 1, 9, 1, 8, 4, 10, 1, 14, 1, 16, 5, 16, 1, 24, 2, 22, 5, 28, 1, 37, 1, 36, 7, 38, 4, 55, 1, 48, 9, 63, 1, 73, 1, 76, 12, 76, 1, 105, 2, 98, 11, 116, 1, 128, 5, 143, 14, 142, 1, 186, 1, 168, 18, 202, 5, 223, 1, 240, 17, 247, 1, 305, 1, 286, 23

Offset: 0

Author

Gus Wiseman, Jan 01 2019

Keywords

• nonn

Comments

First differs from A072721 at a(12) = 9, A072721(12) = 10.

First differs from A379957 at a(16) = 10, A379957(16) = 9.

Examples

			The a(2) = 1 through a(12) = 9 integer partitions (A = 10, B = 11):
  (2)  (3)  (4)   (5)  (6)    (7)  (8)     (9)    (A)      (B)  (66)
            (22)       (33)        (44)    (333)  (55)          (84)
                       (42)        (422)          (82)          (93)
                       (222)       (2222)         (442)         (444)
                                                  (4222)        (822)
                                                  (22222)       (3333)
                                                                (4422)
                                                                (42222)
                                                                (222222)
The a(20) = 16 integer partitions:
  (10,10), (16,4),
  (8,8,4), (16,2,2),
  (5,5,5,5), (8,4,4,4), (8,8,2,2),
  (4,4,4,4,4), (8,4,4,2,2),
  (4,4,4,4,2,2), (8,4,2,2,2,2),
  (4,4,4,2,2,2,2), (8,2,2,2,2,2,2),
  (4,4,2,2,2,2,2,2),
  (4,2,2,2,2,2,2,2,2),
  (2,2,2,2,2,2,2,2,2,2).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to partitions.

Crossrefs

Cf. A001597, A005117, A018819, A023893, A052410, A072720, A072721, A072774, A102430, A322900, A322903, A322911, A322912, A379957.

Programs

Mathematica

radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
powsqfQ[n_]:=SameQ@@Last/@FactorInteger[n];
Table[Length[Select[IntegerPartitions[n],And[FreeQ[#,1],And@@powsqfQ/@#,SameQ@@radbase/@#]&]],{n,30}]

PARI

a(n)={if(n==0, 1, sumdiv(n, d, if(d>1&&issquarefree(d), polcoef(1/prod(j=1, logint(n, d), 1 - x^(d^j), Ser(1, x, 1+n)), n))))} \\ Andrew Howroyd, Jan 23 2025

PARI

seq(n)={Vec(1 + sum(d=2, n, if(issquarefree(d), -1 + 1/prod(j=1, logint(n, d), 1 - x^(d^j), Ser(1, x, 1+n)))))} \\ Andrew Howroyd, Jan 23 2025

Formula

From Andrew Howroyd, Jan 23 2025: (Start)

G.f.: 1 + Sum_{k>=2} -1 + 1/Product_{j>=1} (1 - x^(A005117(k)^j)).

a(p) = 1 for prime p. (End)

Extensions

a(66) onwards from Andrew Howroyd, Jan 23 2025