A279912 a(n) = Sum_{i=1..n} denominator(i^n/n).

0, 1, 3, 7, 10, 21, 21, 43, 36, 57, 63, 111, 70, 157, 129, 147, 136, 273, 171, 343, 210, 301, 333, 507, 252, 505, 471, 495, 430, 813, 441, 931, 528, 777, 819, 903, 570, 1333, 1029, 1099, 756, 1641, 903, 1807, 1110, 1197, 1521, 2163, 952, 2065, 1515, 1911, 1570, 2757

Offset

0,3

Comments

Multiplicative because this sequence is the Dirichlet convolution of A000027 and A127473 which are both multiplicative. - Andrew Howroyd, Aug 05 2018

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Formula

G.f.: Sum_{k>=1} phi(k)^2*x^k/(1 - x^k)^2, where phi(k) is the Euler totient function. - Ilya Gutkovskiy, Jan 02 2017

a(n) = Sum_{k=1..n} gcd(n, k) * phi(n / gcd(n, k)), where phi(k) is the Euler totient function. - Daniel Suteu, Jun 15 2018

a(n) = Sum_{d|n} d * phi(n/d)^2, where phi(k) is the Euler totient function. - Daniel Suteu, Jun 17 2018

Multiplicative with a(p^k) = p^(k-1) * ((p-1) * p^k + 1). - Daniel Suteu, Oct 24 2018

a(n) = Sum_{k=1..n} n/gcd(n,k)*phi(gcd(n,k))^2/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021

Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} (1 - 1/(p*(p+1))) = A065463 / 3 = 0.234814... . - Amiram Eldar, Oct 23 2022

Maple

A279912:=n->add(denom(i^n/n), i=1..n): seq(A279912(n), n=0..100);

Mathematica

Table[DivisorSum[n, # EulerPhi[n/#]^2 &], {n, 53}] (* Michael De Vlieger, Aug 05 2018 *)

Prog

(PARI) a(n) = sum(i=1, n, denominator(i^n/n)); \\ Michel Marcus, Jun 18 2018
(PARI) a(n) = sumdiv(n, d, d*eulerphi(n/d)^2); \\ Michel Marcus, Jun 18 2018
(PARI) a(n) = my(f=factor(n)); if(n==0, 0, prod(k=1, #f~, f[k, 1]^(f[k, 2]-1)  * ((f[k, 1]-1) * f[k, 1]^f[k, 2] + 1))); \\ Daniel Suteu, Oct 24 2018
(GAP) List([0..10], n->Sum([1..n], k->DenominatorRat(k^n/n))); # Muniru A Asiru, Oct 24 2018

Crossrefs

Cf. A000010, A000027, A065463, A127473, A279911.

Sequence in context: A255180 A053159 A345891 * A305477 A316306 A317261

Adjacent sequences: A279909 A279910 A279911 * A279913 A279914 A279915

Keyword

nonn,easy,mult

Author

Wesley Ivan Hurt, Dec 22 2016

Status

approved