cp's OEIS Frontend

A338317 Number of integer partitions of n with no 1's and pairwise coprime distinct parts, where a singleton is always considered coprime.

%I A338317 #12 Nov 05 2020 22:54:55
%S A338317 1,0,1,1,2,2,3,4,5,6,7,11,11,16,16,19,25,32,34,44,46,53,66,80,88,101,
%T A338317 116,132,150,180,204,229,254,287,331,366,426,473,525,584,662,742,835,
%U A338317 922,1013,1128,1262,1408,1555,1711,1894,2080,2297,2555,2806,3064,3376
%N A338317 Number of integer partitions of n with no 1's and pairwise coprime distinct parts, where a singleton is always considered coprime.
%F A338317 The Heinz numbers of these partitions are given by A338316. 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 A338317 The a(2) = 1 through a(12) = 11 partitions (A = 10, B = 11, C = 12):
%e A338317   2   3   4    5    6     7     8      9      A       B       C
%e A338317           22   32   33    43    44     54     55      65      66
%e A338317                     222   52    53     72     73      74      75
%e A338317                           322   332    333    433     83      444
%e A338317                                 2222   522    532     92      543
%e A338317                                        3222   3322    443     552
%e A338317                                               22222   533     732
%e A338317                                                       722     3333
%e A338317                                                       3332    5322
%e A338317                                                       5222    33222
%e A338317                                                       32222   222222
%t A338317 Table[Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&(SameQ@@#||CoprimeQ@@Union[#])&]],{n,0,15}]
%Y A338317 A007359 (A302568) gives the strict case.
%Y A338317 A101268 (A335235) gives pairwise coprime or singleton compositions.
%Y A338317 A200976 (A338318) gives the pairwise non-coprime instead of coprime version.
%Y A338317 A304709 (A304711) gives partitions whose distinct parts are pairwise coprime, with strict case A305713 (A302797).
%Y A338317 A304712 (A338331) allows 1's, with strict version A007360 (A302798).
%Y A338317 A327516 (A302696) gives pairwise coprime partitions.
%Y A338317 A328673 (A328867) gives partitions with no distinct relatively prime parts.
%Y A338317 A338315 (A337987) does not consider singletons coprime.
%Y A338317 A338317 (A338316) gives these partitions.
%Y A338317 A337462 (A333227) gives pairwise coprime compositions.
%Y A338317 A337485 (A337984) gives pairwise coprime integer partitions with no 1's.
%Y A338317 A337665 (A333228) gives compositions with pairwise coprime distinct parts.
%Y A338317 A337667 (A337666) gives pairwise non-coprime compositions.
%Y A338317 A337697 (A022340 /\ A333227) = pairwise coprime compositions with no 1's.
%Y A338317 A337983 (A337696) gives pairwise non-coprime strict compositions, with unordered version A318717 (A318719).
%Y A338317 Cf. A051424, A289508, A302569, A303140, A337694.
%K A338317 nonn
%O A338317 0,5
%A A338317 _Gus Wiseman_, Oct 24 2020