A379308 - OEIS

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

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OFFSET

0,6

LINKS

Table of n, a(n) for n=0..60.

EXAMPLE

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)

MATHEMATICA

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

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.

Sequence in context: A124759 A071295 A296062 * A214178 A117652 A103223

Adjacent sequences: A379305 A379306 A379307 * A379309 A379310 A379311

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 26 2024

STATUS

approved