cp's OEIS Frontend

A331382 Numbers whose sum of prime factors is divisible by their product of prime indices.

%I A331382 #4 Jan 17 2020 10:37:59
%S A331382 1,2,4,8,16,18,20,32,35,44,60,62,64,65,68,72,92,95,98,128,154,160,168,
%T A331382 256,264,288,291,303,324,364,400,476,480,512,618,623,624,642,706,763,
%U A331382 791,812,816,826,938,994,1024,1036,1064,1068,1106,1144,1148,1152,1162
%N A331382 Numbers whose sum of prime factors is divisible by their product of prime indices.
%C A331382 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 A331382 The sequence of terms together with their prime indices begins:
%e A331382      1: {}
%e A331382      2: {1}
%e A331382      4: {1,1}
%e A331382      8: {1,1,1}
%e A331382     16: {1,1,1,1}
%e A331382     18: {1,2,2}
%e A331382     20: {1,1,3}
%e A331382     32: {1,1,1,1,1}
%e A331382     35: {3,4}
%e A331382     44: {1,1,5}
%e A331382     60: {1,1,2,3}
%e A331382     62: {1,11}
%e A331382     64: {1,1,1,1,1,1}
%e A331382     65: {3,6}
%e A331382     68: {1,1,7}
%e A331382     72: {1,1,1,2,2}
%e A331382     92: {1,1,9}
%e A331382     95: {3,8}
%e A331382     98: {1,4,4}
%e A331382    128: {1,1,1,1,1,1,1}
%e A331382 For example, 60 has prime factors {2,2,3,5} and prime indices {1,1,2,3}, and 12 is divisible by 6, so 60 is in the sequence.
%t A331382 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A331382 Select[Range[100],Divisible[Plus@@Prime/@primeMS[#],Times@@primeMS[#]]&]
%Y A331382 These are the Heinz numbers of the partitions counted by A331381.
%Y A331382 Numbers divisible by the sum of their prime factors are A036844.
%Y A331382 Partitions whose product is divisible by their sum are A057568.
%Y A331382 Numbers divisible by the sum of their prime indices are A324851.
%Y A331382 Product of prime indices is divisible by sum of prime indices: A326149.
%Y A331382 Partitions whose Heinz number is divisible by their sum are A330950.
%Y A331382 Sum of prime factors is divisible by sum of prime indices: A331380
%Y A331382 Partitions whose product is equal to the sum of primes are A331383.
%Y A331382 Product of prime indices equals sum of prime factors: A331384.
%Y A331382 Cf. A000040, A001414, A324850, A330953, A330954, A331378, A331379.
%K A331382 nonn
%O A331382 1,2
%A A331382 _Gus Wiseman_, Jan 16 2020