OFFSET
1,2
COMMENTS
a(n) is the sum of all pairs of greater common divisors for (i,j) where 1 <= i <= j <= n. - Jianing Song, Feb 07 2021
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Olivier Bordellès, A note on the average order of the gcd-sum function, Journal of Integer Sequences, vol 10 (2007), article 07.3.3.
FORMULA
According to Bordellès (2007), a(n) = (n^2 / (2*zeta(2)))*(log n + gamma - 1/2 + log(A^12/(2*Pi))) + O(n^(1+theta+epsilon)) where gamma = A001620, A ~= 1.282427129 is the Glaisher-Kinkelin constant A074962, theta is a certain constant defined in terms of the divisor function and known to lie between 1/4 and 131/416, and epsilon is any positive number.
G.f.: (1/(1 - x))*Sum_{k>=1} phi(k)*x^k/(1 - x^k)^2, where phi(k) is the Euler totient function. - Ilya Gutkovskiy, Jan 02 2017
a(n) = (1/2)*Sum_{k=1..n} phi(k) * floor(n/k) * floor(1+n/k), where phi(k) is the Euler totient function. - Daniel Suteu, May 28 2018
From Jianing Song, Feb 07 2021: (Start)
a(n) = Sum_{i=1..n, j=i..n} gcd(i,j).
a(n) = (A018806(n) + n*(n+1)/2) / 2 = (Sum_{k=1..n} phi(k)*(floor(n/k))^2 + n*(n+1)/2) / 2, phi = A000010.
a(n) = A178881(n) + n*(n+1)/2.
EXAMPLE
The gcd-sum function takes values 1,3,5 for n=1,2,3; therefore a(3) = 1+3+5=9.
MATHEMATICA
b[n_] := GCD[n, #]& /@ Range[n] // Total;
Array[b, 100] // Accumulate (* Jean-François Alcover, Jun 05 2021 *)
PROG
(PARI) first(n)=my(v=vector(n), f); v[1]=1; for(i=2, n, f=factor(i); v[i] = v[i-1] + prod(j=1, #f~, (f[j, 2]*(f[j, 1]-1)/f[j, 1] + 1)*f[j, 1]^f[j, 2])); v \\ Charles R Greathouse IV, May 05 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gareth McCaughan, May 05 2016
STATUS
approved