A351397 Sum of the exponents in the prime factorizations of the prime power divisors of n.

0, 1, 1, 3, 1, 2, 1, 6, 3, 2, 1, 4, 1, 2, 2, 10, 1, 4, 1, 4, 2, 2, 1, 7, 3, 2, 6, 4, 1, 3, 1, 15, 2, 2, 2, 6, 1, 2, 2, 7, 1, 3, 1, 4, 4, 2, 1, 11, 3, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 5, 1, 2, 4, 21, 2, 3, 1, 4, 2, 3, 1, 9, 1, 2, 4, 4, 2, 3, 1, 11, 10, 2, 1, 5, 2, 2, 2, 7, 1, 5, 2, 4, 2

Offset

1,4

Comments

a(n) is the sum of all the k's in the divisors of n of the form p^k, where p is prime and k>=1.

Links

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

Formula

a(n) = Sum_{d|n} Omega(d) * [omega(d) = 1].

Additive with a(p^e) = e*(e+1)/2. - Amiram Eldar, Feb 10 2022

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{k>=2} (k * P(k)) = 2.14822166379843041578..., where P(s) is the prime zeta function. - Amiram Eldar, Oct 05 2023

Example

a(8) = 6; The prime power divisors of 8 are 2,4,8 with prime factorizations 2^1,2^2,2^3 and the sum of the exponents in their prime factorizations is 1+2+3 = 6.
a(20) = 4; The prime power divisors of 20 are 2,4,5 with prime factorizations 2^1,2^2,5^1 and the sum of the exponents in each of their prime factorizations is 1+2+1 = 5.

Mathematica

f[p_, e_] := e*(e + 1)/2; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 10 2022 *)

Prog

(PARI) a(n) = sumdiv(n, d, my(x); if (x=isprimepower(d), x)); \\ Michel Marcus, Feb 10 2022

Crossrefs

Cf. A001221 (omega), A001222 (Omega), A077761, A246655.

Sequence in context: A010123 A039620 A008296 * A140185 A229341 A106790

Adjacent sequences: A351394 A351395 A351396 * A351398 A351399 A351400

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Feb 09 2022

Status

approved