A344997 a(n) = A173557(n) * A344753(n).

0, 2, 4, 5, 8, 24, 12, 11, 14, 64, 20, 56, 24, 120, 144, 23, 32, 78, 36, 152, 264, 280, 44, 120, 44, 384, 44, 288, 56, 672, 60, 47, 600, 640, 624, 182, 72, 792, 816, 328, 80, 1296, 84, 680, 480, 1144, 92, 248, 90, 332, 1344, 936, 104, 240, 1360, 624, 1656, 1792, 116, 1536, 120, 2040, 888, 95, 1824, 3120, 132, 1568, 2376

Offset: 1

Author

Antti Karttunen, Jun 05 2021

Keywords

• nonn

Crossrefs

Cf. A008966, A173557, A344753.

Cf. also A344996.

Cf. A002117, A013661, A307868.

Programs

Mathematica

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); a[1] = 0; a[n_] := Module[{fct = FactorInteger[n], p}, p = fct[[;; , 1]]; Times @@ (p - 1)*(Times @@ f @@@ fct + n*Times @@ (1 + 1/p) - 2*n)]; Array[a, 100] (* Amiram Eldar, Dec 08 2023 *)

PARI

A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A344753(n) = sumdiv(n,d,(dA344997(n) = (A173557(n)*A344753(n));

Formula

a(n) = A173557(n) * A344753(n).

a(n) = Product(p_i - 1) * [Sum_{d|n, dA008966(n/d) * d)], where p_i are distinct primes dividing n.

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 1/zeta(2) - 2 * A307868 + zeta(2)*zeta(3) * Product_{p prime} (1 - 2/p^2 - 1/p^3 + 1/p^4 + 3/p^5 - 2/p^6) = 0.283799589272... . - Amiram Eldar, Dec 08 2023