cp's OEIS Frontend

A334353 Least positive integer m relatively prime to n such that sigma(m*n) is a fourth power, where sigma(k) is the sum of the divisors of k.

%I A334353 #16 Jul 27 2024 20:56:18
%S A334353 1,255,170,3783,102,85,31,39063,34711,51,85,1261,1164,53,34,417067,30,
%T A334353 716125,499,55563,127,345,34,13021,417067,55563,3493,117273,10776,17,
%U A334353 7,34359,230,15,321,10549987,2469230,13021,388,8483,28128,187,5323,30865,314758,17,230,1345225,1481538,9473379,10,291,14,82445,17,60615,1999,7495,5960,18521
%N A334353 Least positive integer m relatively prime to n such that sigma(m*n) is a fourth power, where sigma(k) is the sum of the divisors of k.
%C A334353 Conjecture: For any positive integers k and m, there is a positive integer n relatively prime to m such that sigma(m*n) is a k-th power.
%C A334353 This implies that a(n) exists for every n = 1,2,3,....
%C A334353 See also A334350 for a similar conjecture involving Euler's totient function (A000010).
%H A334353 Zhi-Wei Sun, <a href="/A334353/b334353.txt">Table of n, a(n) for n = 1..127</a>
%e A334353 a(2) = 255 with gcd(2, 255) = 1 and sigma(2*255) = sigma(2)*sigma(255) = 3*432 = 1296 = 6^4.
%e A334353 a(64) = 1851519543 with gcd(64, 1851519543) = 1 and sigma(64*1851519543) = sigma(64)*sigma(1851519543) = 127*2654704368 = 337147454736 = 762^4.
%t A334353 QQ[n_]:=QQ[n]=IntegerQ[n^(1/4)];
%t A334353 sigma[n_]:=sigma[n]=DivisorSigma[1,n];
%t A334353 tab={};Do[m=0;Label[aa];m=m+1;If[GCD[m,n]==1&&QQ[sigma[m]*sigma[n]],tab=Append[tab,m],Goto[aa]],{n,1,60}];tab
%o A334353 (PARI) a(n) = my(m=1,s=sigma(n)); while (!((gcd(n, m) == 1) && ispower(s*sigma(m), 4)), m++); m; \\ _Michel Marcus_, Apr 25 2020
%Y A334353 Cf. A000010, A000203, A000583, A006532, A019422, A020477, A334350.
%K A334353 nonn
%O A334353 1,2
%A A334353 _Zhi-Wei Sun_, Apr 24 2020