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

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, 53, 59, 61, 64, 67, 71, 73, 79, 83, 84, 89, 97, 101, 103, 107, 108, 109, 112, 113, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 173, 179, 181, 191, 192, 193

Offset: 1

Author

Gus Wiseman, Jun 10 2019

Keywords

• nonn

Comments

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.

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).

Examples

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  12: {1,1,2}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  23: {9}
  29: {10}
  30: {1,2,3}
  31: {11}
  32: {1,1,1,1,1}
  37: {12}
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

One and positions of ones in A326153.

Cf. A003963, A056239, A057567, A057568, A112798, A301987.

Cf. A325037, A325042, A325044, A326150, A326151, A326154, A326156, A326158.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Divisible[Plus@@primeMS[#],Times@@primeMS[#]]&]