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%I #6 Mar 18 2019 08:14:32
%S 1,3,5,7,9,11,13,17,19,21,23,25,27,29,31,33,35,37,39,41,43,47,49,51,
%T 53,57,59,61,63,65,67,71,73,77,79,81,83,85,87,89,91,93,95,97,99,101,
%U 103,107,109,111,113,115,117,121,123,125,127,129,131,133,137,139
%N Heinz numbers of integer partitions not containing 1 or any part whose prime indices all belong to the partition.
%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%e The sequence of terms together with their prime indices begins:
%e 1: {}
%e 3: {2}
%e 5: {3}
%e 7: {4}
%e 9: {2,2}
%e 11: {5}
%e 13: {6}
%e 17: {7}
%e 19: {8}
%e 21: {2,4}
%e 23: {9}
%e 25: {3,3}
%e 27: {2,2,2}
%e 29: {10}
%e 31: {11}
%e 33: {2,5}
%e 35: {3,4}
%e 37: {12}
%e 39: {2,6}
%e 41: {13}
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[100],!MemberQ[primeMS[#],k_/;SubsetQ[primeMS[#],primeMS[k]]]&]
%Y The subset version is A324739, with maximal case A324762. The strict integer partition version is A324750. The integer partition version is A324755. An infinite version is A324694.
%Y Cf. A000720, A001221, A007097, A056239, A112798, A289509, A290822, A306844, A324695, A324696, A324737, A324744.
%K nonn
%O 1,2
%A _Gus Wiseman_, Mar 17 2019
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