A379309 Number of strict integer partitions of n with a unique squarefree part.

0, 1, 1, 1, 0, 2, 2, 2, 0, 2, 4, 4, 1, 4, 7, 7, 2, 6, 8, 11, 4, 9, 13, 17, 7, 13, 20, 22, 13, 20, 29, 33, 21, 29, 40, 47, 27, 41, 56, 64, 42, 59, 77, 85, 60, 74, 104, 115, 83, 101, 141, 155, 113, 138, 179, 206, 156, 183, 236, 272, 212, 239, 309, 343, 282, 315

Offset: 0

Author

Gus Wiseman, Dec 27 2024

Keywords

• nonn

Examples

			The a(9) = 2 through a(15) = 7 partitions:
  (5,4)  (10)   (11)   (9,3)  (13)     (14)     (15)
  (8,1)  (6,4)  (7,4)         (8,5)    (8,6)    (8,7)
         (8,2)  (8,3)         (12,1)   (9,5)    (9,6)
         (9,1)  (9,2)         (8,4,1)  (10,4)   (11,4)
                                       (12,2)   (12,3)
                                       (8,4,2)  (8,4,3)
                                       (9,4,1)  (9,4,2)
		

Crossrefs

If all parts are squarefree we have A087188, non-strict A073576 (ranks A302478).

If no parts are squarefree we have A256012, non-strict A114374 (ranks A379307).

For composite instead of squarefree we have A379303, non-strict A379302 (ranks A379301).

For prime instead of squarefree we have A379305, non-strict A379304 (ranks A331915).

The non-strict version is A379308, ranks A379316.

For old prime instead of squarefree we have A379315, non-strict A379314 (ranks A379312).

Ranked by A379316 /\ A005117 = squarefree positions of 1 in A379306.

A000041 counts integer partitions, strict A000009.

A005117 lists the squarefree numbers, differences A076259.

A013929 lists the nonsquarefree numbers, differences A078147.

A377038 gives k-th differences of squarefree numbers.

A379310 counts nonsquarefree prime indices.

Cf. A000586, A000607, A002095, A023895, A034891, A036497, A072284, A073247, A096258, A204389, A377430.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?SquareFreeQ]==1&]],{n,0,30}]

PARI

lista(nn) = my(r=1, s=0); for(k=1, nn, if(issquarefree(k), s+=x^k, r*=1+x^k)); concat(0, Vec(r*s+O(x^(1+nn)))); \\ Jinyuan Wang, Feb 21 2025

Extensions

More terms from Jinyuan Wang, Feb 21 2025