A344997 - OEIS

%I #10 Dec 08 2023 12:44:04

%S 0,2,4,5,8,24,12,11,14,64,20,56,24,120,144,23,32,78,36,152,264,280,44,

%T 120,44,384,44,288,56,672,60,47,600,640,624,182,72,792,816,328,80,

%U 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

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

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

%F a(n) = Product(p_i - 1) * [Sum_{d|n, d<n} d+(A008966(n/d) * d)], where p_i are distinct primes dividing n.

%F 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

%t 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 *)

%o (PARI)

%o A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));

%o A344753(n) = sumdiv(n,d,(d<n)*(d+(issquarefree(n/d) * d)));

%o A344997(n) = (A173557(n)*A344753(n));

%Y Cf. A008966, A173557, A344753.

%Y Cf. also A344996.

%Y Cf. A002117, A013661, A307868.

%K nonn

%O 1,2

%A _Antti Karttunen_, Jun 05 2021