A334350 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).

1, 1, 16, 15, 8, 703, 247, 5, 247, 489, 1255, 5, 109, 247, 4, 3, 1, 247, 73, 3, 109, 1255, 13315, 163, 753, 109, 73, 109, 1373, 163, 27331, 1, 625, 1, 81, 109, 57, 73, 1295, 1, 251, 109, 74663, 625, 949, 13315, 1557377, 1, 74663, 753, 16, 81, 175765, 73, 251, 81, 37, 1373, 243895, 1

Offset

1,3

Comments

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.

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

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

a(n) = 1 if and only if n is in A078164. - Charles R Greathouse IV, Apr 24 2020

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..226

Zhi-Wei Sun, Conjectures on representations involving primes, 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 arXiv, arXiv:1211.1588 [math.NT], 2012-2017.

Example

a(3) = 16 with gcd(3,16) = 1 and phi(3*16) = phi(3)*phi(16) = 2*8 = 2^4.
a(167) = 370517977 with gcd(167, 370517977) = 1 and phi(167*370517977) = phi(167)*phi(370517977) = 166*370517976 = 61505984016 = 498^4.

Mathematica

QQ[n_]:=QQ[n]=IntegerQ[n^(1/4)];
phi[n_]:=phi[n]=EulerPhi[n];
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

Prog

(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

Crossrefs

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

Sequence in context: A072311 A221814 A159454 * A306305 A070541 A022972

Adjacent sequences: A334347 A334348 A334349 * A334351 A334352 A334353

Keyword

nonn

Author

Zhi-Wei Sun, Apr 24 2020

Status

approved