This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A272718 #33 Dec 10 2024 09:58:40
%S A272718 1,4,9,17,26,41,54,74,95,122,143,183,208,247,292,340,373,436,473,545,
%T A272718 610,673,718,818,883,958,1039,1143,1200,1335,1396,1508,1613,1712,1829,
%U A272718 1997,2070,2181,2306,2486,2567,2762,2847,3015,3204,3339,3432,3672,3805,4000,4165
%N A272718 Partial sums of gcd-sum sequence A018804.
%C A272718 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 A272718 Charles R Greathouse IV, <a href="/A272718/b272718.txt">Table of n, a(n) for n = 1..10000</a>
%H A272718 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 A272718 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 A272718 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 A272718 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 A272718 From _Jianing Song_, Feb 07 2021: (Start)
%F A272718 a(n) = Sum_{i=1..n, j=i..n} gcd(i,j).
%F A272718 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 A272718 a(n) = A178881(n) + n*(n+1)/2.
%F A272718 a(n) = A018806(n) - A178881(n). (End)
%e A272718 The gcd-sum function takes values 1,3,5 for n = 1,2,3; therefore a(3) = 1+3+5 = 9.
%t A272718 b[n_] := GCD[n, #]& /@ Range[n] // Total;
%t A272718 Array[b, 100] // Accumulate (* _Jean-François Alcover_, Jun 05 2021 *)
%t A272718 f[p_, e_] := (e*(p - 1)/p + 1)*p^e; s[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[s, 100]] (* _Amiram Eldar_, Dec 10 2024 *)
%o A272718 (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 A272718 Partial sums of A018804.
%Y A272718 Cf. A018806, A000010, A001620, A074962, A178881.
%K A272718 nonn,easy
%O A272718 1,2
%A A272718 _Gareth McCaughan_, May 05 2016