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A328868 Heinz numbers of integer partitions with no two (not necessarily distinct) parts relatively prime, but with no divisor in common to all of the parts.

%I A328868 #5 Nov 01 2019 18:42:06
%S A328868 17719,40807,43381,50431,74269,83143,101543,105703,116143,121307,
%T A328868 123469,139919,140699,142883,171613,181831,185803,191479,203557,
%U A328868 205813,211381,213239,215267,219271,230347,246703,249587,249899,279371,286897,289007,296993,300847
%N A328868 Heinz numbers of integer partitions with no two (not necessarily distinct) parts relatively prime, but with no divisor in common to all of the parts.
%C A328868 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%e A328868 The sequence of terms together with their prime indices begins:
%e A328868    17719: {6,10,15}
%e A328868    40807: {6,14,21}
%e A328868    43381: {6,15,20}
%e A328868    50431: {10,12,15}
%e A328868    74269: {6,10,45}
%e A328868    83143: {10,15,18}
%e A328868   101543: {6,21,28}
%e A328868   105703: {6,15,40}
%e A328868   116143: {12,14,21}
%e A328868   121307: {10,15,24}
%e A328868   123469: {12,15,20}
%e A328868   139919: {6,15,50}
%e A328868   140699: {6,22,33}
%e A328868   142883: {6,10,75}
%e A328868   171613: {6,14,63}
%e A328868   181831: {6,20,45}
%e A328868   185803: {10,14,35}
%e A328868   191479: {14,18,21}
%e A328868   203557: {15,18,20}
%e A328868   205813: {10,15,36}
%e A328868   211381: {10,12,45}
%e A328868   213239: {6,15,70}
%e A328868   215267: {6,10,105}
%e A328868   219271: {6,26,39}
%e A328868   230347: {6,6,10,15}
%t A328868 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A328868 dv=Select[Range[100000],GCD@@primeMS[#]==1&&And[And@@(GCD[##]>1&)@@@Tuples[Union[primeMS[#]],2]]&]
%Y A328868 These are the Heinz numbers of the partitions counted by A202425.
%Y A328868 Terms of A328679 that are not powers of 2.
%Y A328868 The strict case is A318716 (preceded by 2).
%Y A328868 A ranking using binary indices (instead of prime indices) is A326912.
%Y A328868 Heinz numbers of relatively prime partitions are A289509.
%Y A328868 Cf. A000837, A056239, A112798, A200976, A291166, A302796, A316476, A318715, A319752, A319759, A328336, A328672, A328677, A328867.
%K A328868 nonn
%O A328868 1,1
%A A328868 _Gus Wiseman_, Oct 30 2019