A358901 Number of integer partitions of n whose parts have all different numbers of prime factors (A001222).

1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 5, 7, 9, 8, 9, 11, 11, 15, 16, 16, 18, 20, 22, 26, 28, 31, 32, 36, 40, 45, 46, 46, 50, 59, 64, 70, 75, 78, 83, 89, 94, 108, 106, 104, 120, 137, 142, 147, 150, 161, 174, 190, 200, 220, 226, 224, 248, 274, 274, 287, 301, 320, 340, 351, 361

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 101 terms from Lucas A. Brown)

Lucas A. Brown, Python program.

Example

The a(1) = 1 through a(11) = 7 partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)    (B)
            (21)  (31)  (41)  (42)  (43)   (62)   (54)   (82)   (74)
                              (51)  (61)   (71)   (63)   (91)   (65)
                                    (421)  (431)  (81)   (451)  (83)
                                                  (621)  (631)  (92)
                                                                (A1)
                                                                (821)

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@PrimeOmega/@#&]], {n, 0, 60}]

Crossrefs

The weakly decreasing version is A358909 (complement A358910).

The version not counting multiplicity is A358903, weakly decreasing A358902.

For equal numbers of prime factors we have A319169, compositions A358911.

A001222 counts prime factors, distinct A001221.

A063834 counts twice-partitions.

A358836 counts multiset partitions with all distinct block sizes.

Cf. A056239, A129519, A141199, A218482, A300335, A319071, A320324, A358335, A358831, A358908.

Sequence in context: A105098 A303003 A063124 * A070547 A319922 A289139

Adjacent sequences: A358898 A358899 A358900 * A358902 A358903 A358904

Keyword

nonn

Author

Gus Wiseman, Dec 07 2022

Extensions

a(61) and beyond from Lucas A. Brown, Dec 14 2022

Status

approved