A334350 - OEIS

%I #26 Apr 28 2020 21:11:13

%S 1,1,16,15,8,703,247,5,247,489,1255,5,109,247,4,3,1,247,73,3,109,1255,

%T 13315,163,753,109,73,109,1373,163,27331,1,625,1,81,109,57,73,1295,1,

%U 251,109,74663,625,949,13315,1557377,1,74663,753,16,81,175765,73,251,81,37,1373,243895,1

%N Least positive integer m relatively prime to n such that phi(m*n) = phi(m)*phi(n) is a fourth power, where phi is Euler's totient function (A000010).

%C Conjecture: For any positive integers k and m, there is a positive integer n relatively prime to m such that phi(m*n) = phi(m)*phi(n) is a k-th power.

%C This conjecture implies that a(n) exists for every n = 1,2,3,....

%C See also A334353 for a similar conjecture involving the sigma function (A000203).

%C a(n) = 1 if and only if n is in A078164. - _Charles R Greathouse IV_, Apr 24 2020

%H Zhi-Wei Sun, <a href="/A334350/b334350.txt">Table of n, a(n) for n = 1..226</a>

%H Zhi-Wei Sun, <a href="https://doi.org/10.1007/978-3-319-68032-3_20">Conjectures on representations involving primes</a>, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. See also <a href="http://arxiv.org/abs/1211.1588">arXiv</a>, arXiv:1211.1588 [math.NT], 2012-2017.

%e a(3) = 16 with gcd(3,16) = 1 and phi(3*16) = phi(3)*phi(16) = 2*8 = 2^4.

%e a(167) = 370517977 with gcd(167, 370517977) = 1 and phi(167*370517977) = phi(167)*phi(370517977) = 166*370517976 = 61505984016 = 498^4.

%t QQ[n_]:=QQ[n]=IntegerQ[n^(1/4)];

%t phi[n_]:=phi[n]=EulerPhi[n];

%t tab={};Do[m=0;Label[aa];m=m+1;If[GCD[m,n]==1&&QQ[phi[m]*phi[n]],tab=Append[tab,m],Goto[aa]],{n,1,60}];tab

%o (PARI) a(n) = my(m=1,e=eulerphi(n)); while (!((gcd(n, m) == 1) && ispower(e*eulerphi(m), 4)), m++); m; \\ _Michel Marcus_, Apr 25 2020

%Y Cf. A000010, A000203, A000583, A039770, A039771, A259915, A259916, A334337, A280988, A334339, A334353, A078164.

%K nonn

%O 1,3

%A _Zhi-Wei Sun_, Apr 24 2020