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A338315
Number of integer partitions of n with no 1's whose distinct parts are pairwise coprime, where a singleton is not considered coprime unless it is (1).
4
0, 0, 0, 0, 0, 1, 0, 3, 2, 4, 4, 10, 6, 15, 13, 16, 21, 31, 29, 43, 41, 50, 63, 79, 81, 99, 113, 129, 145, 179, 197, 228, 249, 284, 328, 363, 418, 472, 522, 581, 655, 741, 828, 921, 1008, 1123, 1259, 1407, 1546, 1709, 1889, 2077, 2292, 2554, 2799, 3061, 3369
OFFSET
0,8
COMMENTS
The Heinz numbers of these partitions are given by A337987. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
EXAMPLE
The a(5) = 1 through a(13) = 15 partitions (empty column indicated by dot, A = 10, B = 11):
32 . 43 53 54 73 65 75 76
52 332 72 433 74 543 85
322 522 532 83 552 94
3222 3322 92 732 A3
443 5322 B2
533 33222 544
722 553
3332 733
5222 922
32222 4333
5332
7222
33322
52222
322222
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !MemberQ[#, 1]&&CoprimeQ@@Union[#]&]], {n, 0, 30}]
CROSSREFS
A200976 is a pairwise non-coprime instead of pairwise coprime version.
A304709 allows 1's, with strict case A305713 and Heinz numbers A304711.
A318717 counts pairwise non-coprime strict partitions.
A337485 is the strict version, with Heinz numbers A337984.
A337987 gives the Heinz numbers of these partitions.
A338317 considers singletons coprime, with Heinz numbers A338316.
A007359 counts singleton or pairwise coprime partitions with no 1's.
A327516 counts pairwise coprime partitions, ranked by A302696.
A328673 counts partitions with no two distinct parts relatively prime.
A337462 counts pairwise coprime compositions, ranked by A333227.
A337561 counts pairwise coprime strict compositions.
A337665 counts compositions whose distinct parts are pairwise coprime.
A337667 counts pairwise non-coprime compositions, ranked by A337666.
A337697 counts pairwise coprime compositions with no 1's.
Sequence in context: A240538 A326923 A365613 * A240541 A073369 A021759
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 23 2020
STATUS
approved