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%I #6 Jan 17 2020 10:37:45
%S 2,4,8,16,32,33,39,55,64,65,66,74,77,78,86,91,110,128,130,132,154,156,
%T 164,182,188,220,256,260,264,308,312,364,371,411,440,459,512,513,520,
%U 528,530,616,624,636,689,728,746,755,765,766,855,880,906,915,918,1007
%N Numbers whose sum of prime factors is divisible by their sum of prime indices.
%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.
%e The sequence of terms together with their prime indices begins:
%e 2: {1}
%e 4: {1,1}
%e 8: {1,1,1}
%e 16: {1,1,1,1}
%e 32: {1,1,1,1,1}
%e 33: {2,5}
%e 39: {2,6}
%e 55: {3,5}
%e 64: {1,1,1,1,1,1}
%e 65: {3,6}
%e 66: {1,2,5}
%e 74: {1,12}
%e 77: {4,5}
%e 78: {1,2,6}
%e 86: {1,14}
%e 91: {4,6}
%e 110: {1,3,5}
%e 128: {1,1,1,1,1,1,1}
%e 130: {1,3,6}
%e 132: {1,1,2,5}
%e For example, 132 has prime factors {2,2,3,11} and prime indices {1,1,2,5}, and 18 is divisible by 9, so 132 is in the sequence.
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[2,100],Divisible[Plus@@Prime/@primeMS[#],Plus@@primeMS[#]]&]
%Y These are the Heinz numbers of the partitions counted by A331380.
%Y Numbers divisible by the sum of their prime factors are A036844.
%Y Partitions whose product is divisible by their sum are A057568.
%Y Numbers divisible by the sum of their prime indices are A324851.
%Y Product of prime indices is divisible by sum of prime indices: A326149.
%Y Partitions whose Heinz number is divisible by their sum are A330950.
%Y Heinz number is divisible by sum of primes: A330953.
%Y Partitions whose product divides their sum of primes are A331381.
%Y Partitions whose product is equal to their sum of primes are A331383.
%Y Product of prime indices equals sum of prime factors: A331384.
%Y Cf. A000040, A001414, A056239, A330954, A331378, A331379, A331382, A331415, A331416.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jan 16 2020
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