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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).
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%I #14 Mar 11 2021 03:22:19

%S 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,

%T 113,129,145,179,197,228,249,284,328,363,418,472,522,581,655,741,828,

%U 921,1008,1123,1259,1407,1546,1709,1889,2077,2292,2554,2799,3061,3369

%N 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).

%C 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.

%e The a(5) = 1 through a(13) = 15 partitions (empty column indicated by dot, A = 10, B = 11):

%e 32 . 43 53 54 73 65 75 76

%e 52 332 72 433 74 543 85

%e 322 522 532 83 552 94

%e 3222 3322 92 732 A3

%e 443 5322 B2

%e 533 33222 544

%e 722 553

%e 3332 733

%e 5222 922

%e 32222 4333

%e 5332

%e 7222

%e 33322

%e 52222

%e 322222

%t Table[Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&CoprimeQ@@Union[#]&]],{n,0,30}]

%Y A200976 is a pairwise non-coprime instead of pairwise coprime version.

%Y A304709 allows 1's, with strict case A305713 and Heinz numbers A304711.

%Y A318717 counts pairwise non-coprime strict partitions.

%Y A337485 is the strict version, with Heinz numbers A337984.

%Y A337987 gives the Heinz numbers of these partitions.

%Y A338317 considers singletons coprime, with Heinz numbers A338316.

%Y A007359 counts singleton or pairwise coprime partitions with no 1's.

%Y A327516 counts pairwise coprime partitions, ranked by A302696.

%Y A328673 counts partitions with no two distinct parts relatively prime.

%Y A337462 counts pairwise coprime compositions, ranked by A333227.

%Y A337561 counts pairwise coprime strict compositions.

%Y A337665 counts compositions whose distinct parts are pairwise coprime.

%Y A337667 counts pairwise non-coprime compositions, ranked by A337666.

%Y A337697 counts pairwise coprime compositions with no 1's.

%Y Cf. A051424, A101268, A220377, A302568, A328867, A333228, A337563.

%K nonn

%O 0,8

%A _Gus Wiseman_, Oct 23 2020