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A383707 Heinz numbers of maximally refined strict integer partitions.

%I A383707 #19 Jun 10 2025 23:15:46
%S A383707 1,2,3,6,10,14,15,30,42,66,70,78,105,110,182,210,330,390
%N A383707 Heinz numbers of maximally refined strict integer partitions.
%C A383707 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%C A383707 Also squarefree numbers such that every strict partition of a prime index contains a prime index.
%C A383707 Also squarefree numbers such that no prime index is a sum of distinct non prime indices.
%e A383707 The terms together with their prime indices begin:
%e A383707     1: {}
%e A383707     2: {1}
%e A383707     3: {2}
%e A383707     6: {1,2}
%e A383707    10: {1,3}
%e A383707    14: {1,4}
%e A383707    15: {2,3}
%e A383707    30: {1,2,3}
%e A383707    42: {1,2,4}
%e A383707    66: {1,2,5}
%e A383707    70: {1,3,4}
%e A383707    78: {1,2,6}
%e A383707   105: {2,3,4}
%e A383707   110: {1,3,5}
%e A383707   182: {1,4,6}
%e A383707   210: {1,2,3,4}
%e A383707   330: {1,2,3,5}
%e A383707   390: {1,2,3,6}
%t A383707 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A383707 nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
%t A383707 Select[Range[30],SquareFreeQ[#]&&With[{y=prix[#]},Intersection[y,Total/@nonsets[y]]=={}]&]
%Y A383707 Partitions of this type are counted by A179009.
%Y A383707 Appears to be positions of 1 in A383706.
%Y A383707 For distinct prime indices see A384320.
%Y A383707 The proper version appears to be A384390.
%Y A383707 The conjugate version is A384723.
%Y A383707 A055396 gives least prime index, greatest A061395.
%Y A383707 A056239 adds up prime indices, row sums of A112798.
%Y A383707 Cf. A048767, A130091, A299200, A351294, A381432, A351295, A357982, A381454, A382525, A384321, A384349, A384389.
%K A383707 nonn,more
%O A383707 1,2
%A A383707 _Gus Wiseman_, May 15 2025