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 A122410 #27 Jun 21 2025 11:50:09
%S A122410 0,1,1,3,1,3,1,7,4,5,1,8,1,7,6,15,1,10,1,14,8,11,1,18,6,13,13,20,1,14,
%T A122410 1,31,12,17,10,26,1,19,14,32,1,20,1,32,22,23,1,38,8,26,18,38,1,31,14,
%U A122410 46,20,29,1,36,1,31,30,63,16,32,1,50,24,34,1,58,1,37,32,56,16,38,1,68,40
%N A122410 a(n) is the sum of j's for those k's, 1 <= k <= n, where gcd(k,n) = p^j, p = prime.
%H A122410 Antti Karttunen, <a href="/A122410/b122410.txt">Table of n, a(n) for n = 1..65537</a>
%F A122410 From _Ridouane Oudra_, Jun 06 2025: (Start)
%F A122410 a(p^m) = (p^m-1)/(p-1), for p prime and m >= 0.
%F A122410 a(n) = Sum_{p|n, p prime} phi(n/p), for n a squarefree.
%F A122410 More generally, for all n we have:
%F A122410 a(n) = Sum_{d|n, d is a prime power} A100995(d)*phi(n/d).
%F A122410 a(n) = Sum_{p|n, p prime} ((p^v(n,p)-1)/(p-1))*phi(n/p^v(n,p)), where p^v(n,p) is the highest power of p dividing n.
%F A122410 a(n) = phi(n) * Sum_{p|n, p prime} p*(1-p^(-v(n,p)))/((1-p)^2). (End)
%F A122410 Sum_{k=1..n} a(k) ~ c * zeta(2) * n^2 / 2, where c = Sum_{p prime} (p^2/(p^2-1)^2) = 0.68073222355480674093... . - _Amiram Eldar_, Jun 21 2025
%e A122410 The positive integers k, k <= 12, where gcd(k,12) = a power of a prime, are 1, 2, 3, 4, 8, 9 and 10; gcd(1,12) = p^0, gcd(2,12) = 2^1, gcd(3,12) = 3^1, gcd(4,12) = 2^2, gcd(8,12) = 2^2, gcd(9,12) = 3^1 and gcd(10,12) = 2^1. The sum of the exponents raising the primes is 0+1+1+2+2+1+1 = 8. So a(12) = 8.
%t A122410 f[n_] := Plus @@ Last /@ Flatten[Select[FactorInteger[GCD[Range[n], n]], Length[ # ] == 1 &], 1]; Table[f[n], {n, 80}] (* _Ray Chandler_, Sep 06 2006 *)
%t A122410 a[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; n * Times @@ (1-1/p) * Total[p*(1-1/p^e)/(p - 1)^2]]; a[1] = 0; Array[a, 100] (* _Amiram Eldar_, Jun 21 2025 *)
%o A122410 (PARI) A122410(n) = sum(k=1,n,isprimepower(gcd(n,k))); \\ _Antti Karttunen_, Feb 25 2018
%o A122410 (PARI) a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); eulerphi(f) * sum(i = 1, #p, p[i] * (1 - 1/p[i]^e[i]) / (p[i] - 1)^2);} \\ _Amiram Eldar_, Jun 21 2025
%Y A122410 Cf. A000010, A013661, A100995, A116512, A122411.
%K A122410 nonn
%O A122410 1,4
%A A122410 _Leroy Quet_, Sep 02 2006
%E A122410 Extended by _Ray Chandler_, Sep 06 2006