login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A344997
a(n) = A173557(n) * A344753(n).
3
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
1,2
FORMULA
a(n) = A173557(n) * A344753(n).
a(n) = Product(p_i - 1) * [Sum_{d|n, d<n} d+(A008966(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
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 *)
PROG
(PARI)
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A344753(n) = sumdiv(n, d, (d<n)*(d+(issquarefree(n/d) * d)));
A344997(n) = (A173557(n)*A344753(n));
CROSSREFS
Cf. also A344996.
Sequence in context: A240460 A239405 A097684 * A277289 A218347 A171411
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jun 05 2021
STATUS
approved