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 A368737 #28 Jan 15 2024 09:04:29
%S A368737 1,3,9,8,9,27,13,20,45,27,21,72,25,39,81,48,33,135,37,72,117,63,45,
%T A368737 180,65,75,189,104,57,243,61,112,189,99,117,360,73,111,225,180,81,351,
%U A368737 85,168,405,135,93,432,133,195,297,200,105,567,189,260,333,171,117,648,121,183,585,256,225,567,133,264,405,351
%N A368737 a(n) = Sum_{k = 1..n} gcd(3*k, n).
%C A368737 a(n) equals the number of solutions to the congruence 3*x*y == 0 (mod n) for 1 <= x, y <= n.
%H A368737 Seiichi Manyama, <a href="/A368737/b368737.txt">Table of n, a(n) for n = 1..10000</a>
%H A368737 Peter Bala, <a href="/A368737/a368737.pdf">Notes on A368737</a>
%F A368737 a(3*n) = 9*A018804(n); a(3*n+1) = A018804(3*n+1); a(3*n+2) = A018804(3*n+2).
%F A368737 a(n) = Sum_{d divides n} gcd(3, d)*phi(d)*n/d, where phi(n) = A000010(n)
%F A368737 Multiplicative: a(3^k) = (2*k + 1)*3^k and for prime p not equal to 3, a(p^k) = (k + 1)*p^k - k*p^(k-1).
%F A368737 Define D(n) = Sum_{d divides n} a(d). Then
%F A368737 D(3*n+1) = (3*n + 1)*tau(3*n+1) and D(3*n+2) = (3*n + 2)*tau(3*n+2), where tau(n) = A000005(n), the number of divisors of n.
%F A368737 The sequence {(1/9)*( D(3*n) - D(n) ) : n >= 1} begins {1, 4, 5, 12, 10, 20, 14, 32, 21, 40, 22, 60, 26, 56, 50, 80, 34, 84, 38, 120, 70, 88, ...} and appears to be multiplicative.
%F A368737 Dirichlet g.f.: (1 + 3/3^s)/(1 - 1/3^s) * zeta(s-1)^2/zeta(s).
%F A368737 Sum_{k=1..n} a(k) ~ 9*n^2 * (log(n)/2 - 1/4 + gamma - 3*log(3)/16 - 3*zeta'(2)/Pi^2) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Jan 11 2024
%e A368737 a(6) = 27: each of the 36 pairs (x, y), 1 <= x, y <= 6, is a solution to the congruence 3*x*y == 0 (mod 6) except for the 9 pairs (x, y) with both x and y odd.
%p A368737 seq(add(gcd(3*k, n), k = 1..n), n = 1..70);
%p A368737 # alternative faster program for large n
%p A368737 with(numtheory): seq(add(gcd(3,d)*phi(d)*n/d, d in divisors(n)), n = 1..70);
%t A368737 Table[Sum[GCD[3*k, n], {k, 1, n}], {n, 1, 100}] (* _Vaclav Kotesovec_, Jan 11 2024 *)
%o A368737 (PARI) a(n) = sum(k = 1, n, gcd(3*k, n)); \\ _Michel Marcus_, Jan 11 2024
%Y A368737 Cf. A000010, A018804, A344372, A368736 - A368742.
%K A368737 nonn,mult,easy
%O A368737 1,2
%A A368737 _Peter Bala_, Jan 05 2024