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A328679 Heinz numbers of integer partitions with no two distinct parts relatively prime, but with no divisor in common to all of the parts.

%I A328679 #4 Nov 01 2019 18:41:47
%S A328679 1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,17719,32768,
%T A328679 40807,43381,50431,65536,74269,83143,101543,105703,116143,121307,
%U A328679 123469,131072,139919,140699,142883,171613,181831,185803,191479,203557,205813,211381,213239
%N A328679 Heinz numbers of integer partitions with no two distinct parts relatively prime, but with no divisor in common to all of the parts.
%C A328679 Equals the union A000079 and A328868.
%C A328679 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A328679 A partition with no two distinct parts relatively prime is said to be intersecting.
%e A328679 The sequence of terms together with their prime indices begins:
%e A328679       1: {}
%e A328679       2: {1}
%e A328679       4: {1,1}
%e A328679       8: {1,1,1}
%e A328679      16: {1,1,1,1}
%e A328679      32: {1,1,1,1,1}
%e A328679      64: {1,1,1,1,1,1}
%e A328679     128: {1,1,1,1,1,1,1}
%e A328679     256: {1,1,1,1,1,1,1,1}
%e A328679     512: {1,1,1,1,1,1,1,1,1}
%e A328679    1024: {1,1,1,1,1,1,1,1,1,1}
%e A328679    2048: {1,1,1,1,1,1,1,1,1,1,1}
%e A328679    4096: {1,1,1,1,1,1,1,1,1,1,1,1}
%e A328679    8192: {1,1,1,1,1,1,1,1,1,1,1,1,1}
%e A328679   16384: {1,1,1,1,1,1,1,1,1,1,1,1,1,1}
%e A328679   17719: {6,10,15}
%e A328679   32768: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
%e A328679   40807: {6,14,21}
%e A328679   43381: {6,15,20}
%e A328679   50431: {10,12,15}
%e A328679   65536: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
%t A328679 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A328679 Select[Range[10000],#==1||GCD@@primeMS[#]==1&&And[And@@(GCD[##]>1&)@@@Subsets[Union[primeMS[#]],{2}]]&]
%Y A328679 These are the Heinz numbers of the partitions counted by A328672.
%Y A328679 Terms that are not powers of 2 are A328868.
%Y A328679 The strict case is A318716.
%Y A328679 The version without global relative primality is A328867.
%Y A328679 A ranking using binary indices (instead of prime indices) is A326912.
%Y A328679 The version for non-isomorphic multiset partitions is A319759.
%Y A328679 The version for divisibility (instead of relative primality) is A328677.
%Y A328679 Heinz numbers of relatively prime partitions are A289509.
%Y A328679 Cf. A000837, A056239, A112798, A200976, A202425, A289509, A291166, A302796, A316476, A318715, A319752, A328336.
%K A328679 nonn
%O A328679 1,2
%A A328679 _Gus Wiseman_, Oct 30 2019