A379303 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,7

LINKS

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

EXAMPLE

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)

MATHEMATICA

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

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.

Sequence in context: A127779 A207634 A222862 * A101437 A039856 A301703

Adjacent sequences: A379300 A379301 A379302 * A379304 A379305 A379306

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 25 2024

STATUS

approved