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 A078540 #20 Aug 29 2024 09:16:26
%S A078540 22,38,46,295,235,749,3687,6128,1415,4254,10451,8351,334,4511,3398,
%T A078540 1286,148870,11015,35519,10239,14072,76088,5991,718,11654,30761,7431,
%U A078540 20993,700654,22169,5095,4198,27415,26744,14318,48368,180878,16991,173123,4166,5033,7246
%N A078540 Least non-balanced x (i.e., not in A020492) such that sigma(p(n),x)/phi(x) is an integer, where p(n) = n-th prime.
%H A078540 Amiram Eldar, <a href="/A078540/b078540.txt">Table of n, a(n) for n = 1..129</a>
%F A078540 a(n) = min{x; A000203(x) mod A000005(x) = 0 but sigma(A000040(n), x) mod phi(x) is not 0}.
%e A078540 n=6: prime(6)=13, cases of sigma(13,x)/phi(x) is an integer are listed in A015771: 1, 2, 3, 6, 12, etc.; the first term which is non-balanced, i.e., not in A020492, is a(6) = 749 = A020492(31); a(29) = 700854 and a(45) = 510759 are remarkably large.
%t A078540 Table[fl=1; Do[s1=DivisorSigma[1, n]/EulerPhi[n]; sk=DivisorSigma[Prime[k], n]/EulerPhi[n]; If[ !IntegerQ[s1]&&IntegerQ[sk]&&Equal[fl, 1], Print[{n, Prime[k]}]; fl=0], {n, 1, 1000000}], {k, 1, 100}]
%o A078540 (PARI) lista(nmax) = {my(ps = primes(nmax), pmax = ps[#ps], v = vector(pmax), c = 0, k = 2, f, e, p); while(c < nmax, f = factor(k); e = eulerphi(f); if(sigma(f) % e > 0, for(i = 1, nmax, p = ps[i]; if(!(sigma(f, p) % e) && v[p] == 0, c++; v[p] = k))); k++); for(i = 1, pmax, if(v[i] > 0, print1(v[i], ", ")));} \\ _Amiram Eldar_, Aug 29 2024
%Y A078540 Cf. A000005, A000010, A000040, A015759-A015774, A020492, A078538, A078539.
%K A078540 nonn
%O A078540 1,1
%A A078540 _Labos Elemer_, Dec 02 2002
%E A078540 a(18) corrected and more terms added by _Amiram Eldar_, Aug 29 2024