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

1,2

Comments

First differs from A106492 for n=64.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

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.

Formula

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A008474, A001694, A106492, A338039.

Sequence in context: A029908 A081758 A106492 * A112264 A118503 A086295

Adjacent sequences: A338035 A338036 A338037 * A338039 A338040 A338041

Keyword

nonn

Author

Michel Marcus, Oct 08 2020

Status

approved