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
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1000
Programs
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GAP
List([0..10],n->Sum([1..n],k->DenominatorRat(k^n/n))); # Muniru A Asiru, Oct 24 2018
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Maple
A279912:=n->add(denom(i^n/n), i=1..n): seq(A279912(n), n=0..100);
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Mathematica
Table[DivisorSum[n, # EulerPhi[n/#]^2 &], {n, 53}] (* Michael De Vlieger, Aug 05 2018 *) -
PARI
a(n) = sum(i=1, n, denominator(i^n/n)); \\ Michel Marcus, Jun 18 2018
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PARI
a(n) = sumdiv(n, d, d*eulerphi(n/d)^2); \\ Michel Marcus, Jun 18 2018
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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
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
Comments