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A325780 Heinz numbers of perfect integer partitions.

%I A325780 #6 May 22 2019 21:00:39
%S A325780 1,2,4,6,8,16,18,20,32,42,54,56,64,100,128,162,176,234,256,260,294,
%T A325780 392,416,486,500,512,798,1024,1026,1064,1088,1458,1936,2048,2058,2300,
%U A325780 2432,2500,2744,3042,3380,4096,4374,4698,5104,5408,5888,8192,8658,9620,10878
%N A325780 Heinz numbers of perfect integer partitions.
%C A325780 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A325780 The sum of prime indices of n is A056239(n). A number is in this sequence iff all of its divisors have distinct sums of prime indices, and these sums cover an initial interval of nonnegative integers. For example, the divisors of 260 are {1, 2, 4, 5, 10, 13, 20, 26, 52, 65, 130, 260}, with respective sums of prime indices {0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 11}, so 260 is in the sequence.
%F A325780 Intersection of A299702 (knapsack partitions) and A325781 (complete partitions).
%e A325780 The sequence of terms together with their prime indices begins:
%e A325780       1: {}
%e A325780       2: {1}
%e A325780       4: {1,1}
%e A325780       6: {1,2}
%e A325780       8: {1,1,1}
%e A325780      16: {1,1,1,1}
%e A325780      18: {1,2,2}
%e A325780      20: {1,1,3}
%e A325780      32: {1,1,1,1,1}
%e A325780      42: {1,2,4}
%e A325780      54: {1,2,2,2}
%e A325780      56: {1,1,1,4}
%e A325780      64: {1,1,1,1,1,1}
%e A325780     100: {1,1,3,3}
%e A325780     128: {1,1,1,1,1,1,1}
%e A325780     162: {1,2,2,2,2}
%e A325780     176: {1,1,1,1,5}
%e A325780     234: {1,2,2,6}
%e A325780     256: {1,1,1,1,1,1,1,1}
%e A325780     260: {1,1,3,6}
%t A325780 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A325780 Select[Range[1000],Sort[hwt/@Rest[Divisors[#]]]==Range[DivisorSigma[0,#]-1]&]
%Y A325780 Equals the sorted concatenation of the triangle A258119.
%Y A325780 A subsequence of A299702 and A325781.
%Y A325780 Cf. A002033, A056239, A103300, A108917, A112798, A126796, A188431, A325763, A325768, A325782.
%K A325780 nonn
%O A325780 1,2
%A A325780 _Gus Wiseman_, May 21 2019