A379302 Number of integer partitions of n with a unique composite part.

0, 0, 0, 0, 1, 1, 3, 4, 7, 11, 16, 23, 32, 43, 58, 77, 100, 129, 164, 207, 259, 323, 398, 489, 595, 723, 872, 1049, 1255, 1495, 1774, 2097, 2472, 2903, 3399, 3969, 4618, 5362, 6210, 7173, 8268, 9506, 10907, 12488, 14271, 16278, 18532, 21061, 23893, 27064

Offset: 0

Author

Gus Wiseman, Dec 25 2024

Keywords

• nonn

Examples

			The a(0) = 0 through a(9) = 11 partitions:
  .  .  .  .  (4)  (41)  (6)    (43)    (8)      (9)
                         (42)   (61)    (62)     (54)
                         (411)  (421)   (422)    (63)
                                (4111)  (431)    (81)
                                        (611)    (432)
                                        (4211)   (621)
                                        (41111)  (4221)
                                                 (4311)
                                                 (6111)
                                                 (42111)
                                                 (411111)
		

Crossrefs

If all parts are composite we have A023895 (strict A204389), ranks A320629.

If no parts are composite we have A034891 (strict A036497), ranks A302540.

Ranked by A379301 = positions of 1 in A379300.

The strict case is A379303.

For a unique prime part we have A379304 (strict A379305), ranks A331915.

A000041 counts integer partitions, strict A000009.

A002808 lists the composite numbers, nonprimes A018252.

A066247 is the characteristic function for the composite numbers.

A377033 gives k-th differences of composite numbers.

Cf. A000070, A000586, A000607, A002095, A038348, A096258, A114374, A330944, A379308, A379309, A379314, A379315.

Programs

Mathematica

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