A338038 a(n) is the sum of the primes and exponents in the prime factorization of n, but ignoring 1-exponents.

0, 2, 3, 4, 5, 5, 7, 5, 5, 7, 11, 7, 13, 9, 8, 6, 17, 7, 19, 9, 10, 13, 23, 8, 7, 15, 6, 11, 29, 10, 31, 7, 14, 19, 12, 9, 37, 21, 16, 10, 41, 12, 43, 15, 10, 25, 47, 9, 9, 9, 20, 17, 53, 8, 16, 12, 22, 31, 59, 12, 61, 33, 12, 8, 18, 16, 67, 21, 26, 14, 71, 10

Offset: 1

Author

Michel Marcus, Oct 08 2020

Keywords

• nonn

Comments

First differs from A106492 for n=64.

Examples

			For n = 18 = 2*3^2, a(18) = 2 + (3+2) = 7.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Chris Bispels, Muhammet Boran, Steven J. Miller, Eliel Sosis, and Daniel Tsai, v-Palindromes: An Analogy to the Palindromes, arXiv:2405.05267 [math.HO], 2024.
• Daniel Tsai, A recurring pattern in natural numbers of a certain property, arXiv:2010.03151 [math.NT], 2020.
• Daniel Tsai, A recurring pattern in natural numbers of a certain property, Integers (2021) Vol. 21, Article #A32.

Crossrefs

Cf. A008474, A001694, A106492, A338039.

Programs

Maple

f:= proc(n) local t;
  add(t[1]+t[2],t=subs(1=0,ifactors(n)[2]));
end proc:
map(f, [$1..100]); # Robert Israel, Oct 13 2020

Mathematica

a[1] = 0; a[n_] := Plus @@ First /@ (f = FactorInteger[n]) + Plus @@ Select[Last /@ f, # > 1 &]; Array[a, 100] (* Amiram Eldar, Oct 08 2020 *)

PARI

a(n) = my(f=factor(n)); vecsum(f[,1]) + sum(k=1, #f~, if (f[k,2]!=1, f[k,2]));

Formula

a(n) = A008474(n) for powerful numbers (A001694).