A379308 Number of integer partitions of n with a unique squarefree part.

0, 1, 1, 1, 0, 2, 2, 2, 0, 3, 5, 5, 1, 6, 9, 9, 2, 10, 14, 18, 6, 18, 24, 30, 11, 28, 39, 47, 24, 48, 63, 76, 41, 74, 95, 118, 65, 120, 149, 181, 107, 181, 221, 266, 169, 266, 335, 398, 262, 394, 487, 578, 391, 578, 697, 844, 592, 834, 997, 1198, 867

Offset: 0

Author

Gus Wiseman, Dec 26 2024

Keywords

• nonn

Examples

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

Crossrefs

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

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

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

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

The strict case is A379309.

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

Ranked by A379316, 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, A013928, A023895, A034891, A072284, A073247, A120327, A175804, A376657, A377430.

Programs

Mathematica

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