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A325792 Positive integers with as many proper divisors as the sum of their prime indices.

%I A325792 #7 May 23 2019 14:52:44
%S A325792 1,2,4,6,8,16,18,20,32,42,54,56,64,100,128,162,176,204,234,256,260,
%T A325792 294,308,315,350,392,416,486,500,512,690,696,798,920,1024,1026,1064,
%U A325792 1088,1116,1122,1190,1365,1430,1458,1496,1755,1936,1968,2025,2048,2058,2079
%N A325792 Positive integers with as many proper divisors as the sum of their prime indices.
%C A325792 First differs from A325780 in having 204.
%C A325792 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, with sum A056239(n).
%e A325792 The term 42 is in the sequence because it has 7 proper divisors (1, 2, 3, 6, 7, 14, 21) and its sum of prime indices is also 1 + 2 + 4 = 7.
%e A325792 The sequence of terms together with their prime indices begins:
%e A325792      1: {}
%e A325792      2: {1}
%e A325792      4: {1,1}
%e A325792      6: {1,2}
%e A325792      8: {1,1,1}
%e A325792     16: {1,1,1,1}
%e A325792     18: {1,2,2}
%e A325792     20: {1,1,3}
%e A325792     32: {1,1,1,1,1}
%e A325792     42: {1,2,4}
%e A325792     54: {1,2,2,2}
%e A325792     56: {1,1,1,4}
%e A325792     64: {1,1,1,1,1,1}
%e A325792    100: {1,1,3,3}
%e A325792    128: {1,1,1,1,1,1,1}
%e A325792    162: {1,2,2,2,2}
%e A325792    176: {1,1,1,1,5}
%e A325792    204: {1,1,2,7}
%e A325792    234: {1,2,2,6}
%e A325792    256: {1,1,1,1,1,1,1,1}
%t A325792 Select[Range[100],DivisorSigma[0,#]-1==Total[Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]]&]
%Y A325792 Positions of 1's in A325794.
%Y A325792 Heinz numbers of the partitions counted by A325828.
%Y A325792 Cf. A000005, A002033, A056239, A112798, A299702, A304793.
%Y A325792 Cf. A325694, A325780, A325781, A325793, A325795, A325796, A325797, A325798.
%K A325792 nonn
%O A325792 1,2
%A A325792 _Gus Wiseman_, May 23 2019