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%I #6 Sep 19 2018 09:16:42
%S 1,2,3,5,7,11,13,15,17,19,23,29,31,33,35,37,41,43,47,51,53,55,59,61,
%T 67,69,71,73,77,79,83,85,89,91,93,95,97,101,103,107,109,113,119,123,
%U 127,131,137,139,141,143,145,149,151,155,157,161,163,167,173,177
%N Heinz numbers of integer partitions such that every distinct submultiset has a different GCD.
%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C First differs from A304713 (Heinz numbers of pairwise indivisible partitions) at A304713(58) = 165, which is absent from this sequence because its prime indices are {2,3,5} and GCD(2,3) = GCD(2,3,5) = 1. The first term with more than two prime factors is 17719, which has prime indices {6,10,15}. The first term with more than two prime factors that is absent from A318716 is 296851, which has prime indices {12,20,30}.
%e The sequence of partitions whose Heinz numbers are in the sequence begins: (), (1), (2), (3), (4), (5), (6), (3,2), (7), (8), (9), (10), (11), (5,2), (4,3), (12), (13), (14), (15), (7,2), (16), (5,3).
%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[100],UnsameQ@@GCD@@@Union[Subsets[primeMS[#]]]&]
%Y Cf. A056239, A108917, A122768, A275972, A299702, A301899, A301900, A304713, A316313, A319315, A319318, A319327.
%K nonn
%O 1,2
%A _Gus Wiseman_, Sep 17 2018