A279912 - OEIS

%I #41 Oct 23 2022 02:53:50

%S 0,1,3,7,10,21,21,43,36,57,63,111,70,157,129,147,136,273,171,343,210,

%T 301,333,507,252,505,471,495,430,813,441,931,528,777,819,903,570,1333,

%U 1029,1099,756,1641,903,1807,1110,1197,1521,2163,952,2065,1515,1911,1570,2757

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

%C Multiplicative because this sequence is the Dirichlet convolution of A000027 and A127473 which are both multiplicative. - _Andrew Howroyd_, Aug 05 2018

%H Andrew Howroyd, <a href="/A279912/b279912.txt">Table of n, a(n) for n = 0..1000</a>

%F 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

%F 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

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

%F Multiplicative with a(p^k) = p^(k-1) * ((p-1) * p^k + 1). - _Daniel Suteu_, Oct 24 2018

%F 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

%F 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

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

%t Table[DivisorSum[n, # EulerPhi[n/#]^2 &], {n, 53}] (* _Michael De Vlieger_, Aug 05 2018 *)

%o (PARI) a(n) = sum(i=1, n, denominator(i^n/n)); \\ _Michel Marcus_, Jun 18 2018

%o (PARI) a(n) = sumdiv(n, d, d*eulerphi(n/d)^2); \\ _Michel Marcus_, Jun 18 2018

%o (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

%o (GAP) List([0..10],n->Sum([1..n],k->DenominatorRat(k^n/n))); # _Muniru A Asiru_, Oct 24 2018

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

%K nonn,easy,mult

%O 0,3

%A _Wesley Ivan Hurt_, Dec 22 2016