cp's OEIS Frontend

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).

%I A338315 #14 Mar 11 2021 03:22:19
%S A338315 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 A338315 113,129,145,179,197,228,249,284,328,363,418,472,522,581,655,741,828,
%U A338315 921,1008,1123,1259,1407,1546,1709,1889,2077,2292,2554,2799,3061,3369
%N 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).
%C A338315 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 A338315 The a(5) = 1 through a(13) = 15 partitions (empty column indicated by dot, A = 10, B = 11):
%e A338315   32   .  43    53    54     73     65      75      76
%e A338315           52    332   72     433    74      543     85
%e A338315           322         522    532    83      552     94
%e A338315                       3222   3322   92      732     A3
%e A338315                                     443     5322    B2
%e A338315                                     533     33222   544
%e A338315                                     722             553
%e A338315                                     3332            733
%e A338315                                     5222            922
%e A338315                                     32222           4333
%e A338315                                                     5332
%e A338315                                                     7222
%e A338315                                                     33322
%e A338315                                                     52222
%e A338315                                                     322222
%t A338315 Table[Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&CoprimeQ@@Union[#]&]],{n,0,30}]
%Y A338315 A200976 is a pairwise non-coprime instead of pairwise coprime version.
%Y A338315 A304709 allows 1's, with strict case A305713 and Heinz numbers A304711.
%Y A338315 A318717 counts pairwise non-coprime strict partitions.
%Y A338315 A337485 is the strict version, with Heinz numbers A337984.
%Y A338315 A337987 gives the Heinz numbers of these partitions.
%Y A338315 A338317 considers singletons coprime, with Heinz numbers A338316.
%Y A338315 A007359 counts singleton or pairwise coprime partitions with no 1's.
%Y A338315 A327516 counts pairwise coprime partitions, ranked by A302696.
%Y A338315 A328673 counts partitions with no two distinct parts relatively prime.
%Y A338315 A337462 counts pairwise coprime compositions, ranked by A333227.
%Y A338315 A337561 counts pairwise coprime strict compositions.
%Y A338315 A337665 counts compositions whose distinct parts are pairwise coprime.
%Y A338315 A337667 counts pairwise non-coprime compositions, ranked by A337666.
%Y A338315 A337697 counts pairwise coprime compositions with no 1's.
%Y A338315 Cf. A051424, A101268, A220377, A302568, A328867, A333228, A337563.
%K A338315 nonn
%O A338315 0,8
%A A338315 _Gus Wiseman_, Oct 23 2020