A272718 - OEIS

%I #29 Jun 05 2021 08:43:39

%S 1,4,9,17,26,41,54,74,95,122,143,183,208,247,292,340,373,436,473,545,

%T 610,673,718,818,883,958,1039,1143,1200,1335,1396,1508,1613,1712,1829,

%U 1997,2070,2181,2306,2486,2567,2762,2847,3015,3204,3339,3432,3672,3805,4000,4165

%N Partial sums of gcd-sum sequence A018804.

%C 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

%H Charles R Greathouse IV, <a href="/A272718/b272718.txt">Table of n, a(n) for n = 1..10000</a>

%H Olivier Bordellès, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Bordelles/bordelles90.html">A note on the average order of the gcd-sum function</a>, Journal of Integer Sequences, vol 10 (2007), article 07.3.3.

%F 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.

%F 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

%F 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

%F From _Jianing Song_, Feb 07 2021: (Start)

%F a(n) = Sum_{i=1..n, j=i..n} gcd(i,j).

%F 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.

%F a(n) = A178881(n) + n*(n+1)/2.

%F a(n) = A018806(n) - A178881(n). (End)

%e The gcd-sum function takes values 1,3,5 for n=1,2,3; therefore a(3) = 1+3+5=9.

%t b[n_] := GCD[n, #]& /@ Range[n] // Total;

%t Array[b, 100] // Accumulate (* _Jean-François Alcover_, Jun 05 2021 *)

%o (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

%Y Partial sums of A018804.

%Y Cf. A018806, A000010, A001620, A074962, A178881.

%K nonn,easy

%O 1,2

%A _Gareth McCaughan_, May 05 2016