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

Author

Wesley Ivan Hurt, Feb 09 2022

Keywords

• nonn
• easy

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.

Examples

			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.
		

Links

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

Crossrefs

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

Cf. A090885, A100995.

Programs

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 *)

PARI

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

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

From Ridouane Oudra, Aug 16 2024: (Start)

a(n) = (A001222(n) + A090885(n))/2 ;

a(n) = Sum_{d|n} A100995(d). (End)