A351399 Sum of the exponents in the prime factorizations of the divisors of A100716(n) of the form p^p, p prime.

2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2

Offset

1,1

Comments

The values larger than 0 of the additive function b(n) defined by b(p^e) = p if p <= e, and 0 otherwise. - Amiram Eldar, Oct 05 2023

Links

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

Formula

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (Sum_{p prime} 1/p^(p-1))/(1 - Product_{p prime}(1 - 1/p^p)) = 2.203949957405925045806... . - Amiram Eldar, Oct 05 2023

Example

a(30) = 5; A100716(30) = 108 has two divisors of the form p^p, namely 4 and 27 with prime factorization 2^2 and 3^3. The sum of the exponents of 2^2 and 3^3 is 2+3 = 5.

Mathematica

f[p_, e_] := If[p <= e, p, 0]; b[1] = 0; b[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Array[b, 1000], # > 0 &] (* Amiram Eldar, Oct 05 2023 *)

Prog

(PARI) b(n) = {my(f = factor(n)); sum(i = 1, #f~, if(f[i, 1] <= f[i, 2], f[i, 1], 0))};
lista(max) = {my(b1); for(k =1, max, b1 = b(k); if(b1 > 0, print1(b1, ", "))); } \\ Amiram Eldar, Oct 05 2023

Crossrefs

Cf. A100716.

Sequence in context: A187785 A238277 A258757 * A024708 A096917 A336664

Adjacent sequences: A351396 A351397 A351398 * A351400 A351401 A351402

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Feb 09 2022

Status

approved