A358903 - OEIS

%I #20 Feb 14 2024 09:42:51

%S 1,1,1,2,2,2,2,2,3,4,4,4,4,5,7,8,7,9,10,10,10,9,11,15,14,13,15,14,14,

%T 17,16,17,17,16,16,17,17,21,26,24,23,25,27,29,32,31,29,36,36,35,37,37,

%U 42,49,45,44,50,49,50,58,55,55,58,56,58,66,62,65,75

%N Number of integer partitions of n whose parts have all different numbers of distinct prime factors (A001221).

%H Alois P. Heinz, <a href="/A358903/b358903.txt">Table of n, a(n) for n = 0..10000</a> (first 101 terms from Lucas A. Brown)

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A358903.py">Python program</a>.

%e The a(15) = 8 partitions are: (15), (14,1), (12,3), (12,2,1), (10,5), (10,4,1), (6,9), (8,6,1).

%p p:= proc(n) option remember; nops(ifactors(n)[2]) end:

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,

%p       add((t-> `if`(t<i, b(n-j, t), 0))(p(j)), j=1..n)))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..68);  # _Alois P. Heinz_, Feb 14 2024

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@PrimeNu/@#&]],{n,0,30}]

%Y Counting prime factors with multiplicity gives A358901.

%Y The weakly decreasing version is A358902, with multiplicity A358335.

%Y A001222 counts prime factors, distinct A001221.

%Y A116608 counts partitions by sum and number of distinct parts.

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

%Y Cf. A046660, A071625, A129519, A141199, A319169, A358831, A358909, A358911.

%K nonn

%O 0,4

%A _Gus Wiseman_, Dec 07 2022

%E a(56) and beyond from _Lucas A. Brown_, Dec 14 2022