A379305 Number of strict integer partitions of n with a unique prime part.

0, 0, 1, 2, 1, 1, 2, 3, 3, 3, 3, 6, 8, 8, 8, 10, 12, 17, 18, 18, 22, 28, 30, 36, 40, 44, 52, 62, 67, 78, 87, 97, 113, 129, 137, 156, 177, 200, 227, 251, 271, 312, 350, 382, 425, 475, 521, 588, 648, 705, 785, 876, 957, 1061, 1164, 1272, 1411, 1558, 1693, 1866

Offset: 0

Author

Gus Wiseman, Dec 27 2024

Keywords

• nonn

Examples

			The a(2) = 1 through a(12) = 8 partitions (A=10, B=11):
  (2)  (3)   (31)  (5)  (42)  (7)    (62)   (54)   (82)   (B)    (93)
       (21)             (51)  (43)   (71)   (63)   (541)  (65)   (A2)
                              (421)  (431)  (621)  (631)  (74)   (B1)
                                                          (83)   (642)
                                                          (92)   (651)
                                                          (821)  (741)
                                                                 (831)
                                                                 (921)
		

Crossrefs

For all prime parts we have A000586, non-strict A000607 (ranks A076610).

For no prime parts we have A096258, non-strict A002095 (ranks A320628).

Ranked by A331915 /\ A005117 = squarefree positions of one in A257994.

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

The non-strict version is A379304.

For squarefree instead of prime we have A379309, non-strict A379308 (ranks A379316).

Considering 1 prime gives A379315, non-strict A379314 (ranks A379312).

A000040 lists the prime numbers, differences A001223.

A000041 counts integer partitions, strict A000009.

A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.

A095195 gives k-th differences of prime numbers.

Cf. A000070, A023895, A034891, A036497, A038348, A204389, A302540, A320629, A330944.

Programs

Mathematica

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