A379304 Number of integer partitions of n with a unique prime part.

0, 0, 1, 2, 2, 3, 4, 6, 7, 9, 11, 17, 20, 26, 31, 41, 47, 62, 72, 93, 108, 136, 156, 199, 226, 279, 321, 398, 452, 555, 630, 767, 873, 1051, 1188, 1433, 1618, 1930, 2185, 2595, 2921, 3458, 3891, 4580, 5155, 6036, 6776, 7926, 8883, 10324, 11577, 13421, 15014

Offset: 0

Author

Gus Wiseman, Dec 27 2024

Keywords

• nonn

Examples

			The a(2) = 1 through a(9) = 9 partitions:
  (2)  (3)   (31)   (5)     (42)     (7)       (62)       (54)
       (21)  (211)  (311)   (51)     (43)      (71)       (63)
                    (2111)  (3111)   (421)     (431)      (621)
                            (21111)  (511)     (4211)     (711)
                                     (31111)   (5111)     (4311)
                                     (211111)  (311111)   (42111)
                                               (2111111)  (51111)
                                                          (3111111)
                                                          (21111111)
		

Crossrefs

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

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

Ranked by A331915 = positions of one in A257994.

For a unique composite part we have A379302 (strict A379303), ranks A379301.

The strict case is A379305.

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

Considering 1 prime gives A379314 (strict A379315), 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, A204389, A302540, A330944.

Programs

Mathematica

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