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Number of integer partitions of n whose parts have all different numbers of prime factors (A001222).
10

%I #24 Feb 12 2024 18:26:24

%S 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,

%T 40,45,46,46,50,59,64,70,75,78,83,89,94,108,106,104,120,137,142,147,

%U 150,161,174,190,200,220,226,224,248,274,274,287,301,320,340,351,361

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

%H Alois P. Heinz, <a href="/A358901/b358901.txt">Table of n, a(n) for n = 0..5000</a> (first 101 terms from Lucas A. Brown)

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A358901.py">Python program</a>.

%e The a(1) = 1 through a(11) = 7 partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8) (9) (A) (B)

%e (21) (31) (41) (42) (43) (62) (54) (82) (74)

%e (51) (61) (71) (63) (91) (65)

%e (421) (431) (81) (451) (83)

%e (621) (631) (92)

%e (A1)

%e (821)

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

%Y The weakly decreasing version is A358909 (complement A358910).

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

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

%Y A001222 counts prime factors, distinct A001221.

%Y A063834 counts twice-partitions.

%Y A358836 counts multiset partitions with all distinct block sizes.

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

%K nonn

%O 0,4

%A _Gus Wiseman_, Dec 07 2022

%E a(61) and beyond from _Lucas A. Brown_, Dec 14 2022