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A358903
Number of integer partitions of n whose parts have all different numbers of distinct prime factors (A001221).
7
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, 17, 16, 17, 17, 16, 16, 17, 17, 21, 26, 24, 23, 25, 27, 29, 32, 31, 29, 36, 36, 35, 37, 37, 42, 49, 45, 44, 50, 49, 50, 58, 55, 55, 58, 56, 58, 66, 62, 65, 75
OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 101 terms from Lucas A. Brown)
Lucas A. Brown, Python program.
EXAMPLE
The a(15) = 8 partitions are: (15), (14,1), (12,3), (12,2,1), (10,5), (10,4,1), (6,9), (8,6,1).
MAPLE
p:= proc(n) option remember; nops(ifactors(n)[2]) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
add((t-> `if`(t<i, b(n-j, t), 0))(p(j)), j=1..n)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..68); # Alois P. Heinz, Feb 14 2024
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@PrimeNu/@#&]], {n, 0, 30}]
CROSSREFS
Counting prime factors with multiplicity gives A358901.
The weakly decreasing version is A358902, with multiplicity A358335.
A001222 counts prime factors, distinct A001221.
A116608 counts partitions by sum and number of distinct parts.
A358836 counts multiset partitions with all distinct block sizes.
Sequence in context: A025780 A199121 A109697 * A103373 A038539 A275891
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 07 2022
EXTENSIONS
a(56) and beyond from Lucas A. Brown, Dec 14 2022
STATUS
approved