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A326155 Positive integers whose sum of prime indices is divisible by their product of prime indices.

%I A326155 #10 Jan 11 2021 02:50:17
%S A326155 1,2,3,4,5,7,8,9,11,12,13,16,17,19,23,29,30,31,32,37,40,41,43,47,48,
%T A326155 53,59,61,64,67,71,73,79,83,84,89,97,101,103,107,108,109,112,113,127,
%U A326155 128,131,137,139,144,149,151,157,163,167,173,179,181,191,192,193
%N A326155 Positive integers whose sum of prime indices is divisible by their product of prime indices.
%C A326155 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.
%C A326155 Also Heinz numbers of the integer partitions counted by A057567. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H A326155 Amiram Eldar, <a href="/A326155/b326155.txt">Table of n, a(n) for n = 1..10000</a>
%e A326155 The sequence of terms together with their prime indices begins:
%e A326155    1: {}
%e A326155    2: {1}
%e A326155    3: {2}
%e A326155    4: {1,1}
%e A326155    5: {3}
%e A326155    7: {4}
%e A326155    8: {1,1,1}
%e A326155    9: {2,2}
%e A326155   11: {5}
%e A326155   12: {1,1,2}
%e A326155   13: {6}
%e A326155   16: {1,1,1,1}
%e A326155   17: {7}
%e A326155   19: {8}
%e A326155   23: {9}
%e A326155   29: {10}
%e A326155   30: {1,2,3}
%e A326155   31: {11}
%e A326155   32: {1,1,1,1,1}
%e A326155   37: {12}
%t A326155 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A326155 Select[Range[100],Divisible[Plus@@primeMS[#],Times@@primeMS[#]]&]
%Y A326155 One and positions of ones in A326153.
%Y A326155 Cf. A003963, A056239, A057567, A057568, A112798, A301987.
%Y A326155 Cf. A325037, A325042, A325044, A326150, A326151, A326154, A326156, A326158.
%K A326155 nonn
%O A326155 1,2
%A A326155 _Gus Wiseman_, Jun 10 2019