cp's OEIS Frontend

A331380 Numbers whose sum of prime factors is divisible by their sum of prime indices.

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