A379303 Number of strict integer partitions of n with a unique composite part.

0, 0, 0, 0, 1, 1, 2, 3, 3, 6, 6, 8, 10, 10, 13, 15, 17, 20, 22, 24, 28, 31, 36, 40, 44, 50, 55, 62, 70, 75, 83, 89, 97, 108, 115, 128, 136, 146, 161, 172, 188, 203, 215, 233, 249, 269, 291, 309, 331, 353, 376, 405, 433, 459, 490, 518, 554, 592, 629, 670, 705

Offset: 0

Author

Gus Wiseman, Dec 25 2024

Keywords

• nonn

Examples

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

Crossrefs

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

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

The non-strict version is A379302, ranks A379301 (positions of 1 in A379300).

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

A000040 lists the prime numbers, differences A001223.

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, see A073445, A377034-A377037.

Cf. A000070, A000586, A000607, A002095, A038348, A096258, A113646, A376680, A379308, A379309, A379314, A379315.

Programs

Mathematica

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