A334337 - OEIS

A334337

Least positive integer m such that phi(m*n) is a cube, where phi is Euler's totient function (A000010).

1, 1, 5, 4, 3, 4, 37, 2, 37, 2, 101, 2, 19, 37, 1, 1, 5, 36, 13, 1, 19, 101, 13333, 1, 55, 19, 13, 19, 985, 1, 1057, 4, 401, 4, 73, 18, 7, 13, 9, 4, 275, 18, 2649, 401, 9, 13333, 169285, 4, 1813, 50, 4, 73, 3385, 12, 25, 73, 7, 788, 40371, 4, 3737, 1057, 12, 2, 37, 401, 4357, 2, 6537, 73, 5401, 9, 35, 7, 25, 7, 3737, 9, 48673, 2

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OFFSET

1,3

COMMENTS

Conjecture: a(n) exists for any n > 0. In other words, for any positive integer n, there is a positive integer m with phi(m*n) equal to a cube.

We note that there is no positive integer m <= 10^8 with phi(107*m) equal to a fourth power.

The author's conjecture in A259915 implies that for any positive integer n there is a positive integer m with phi(m*n) equal to a square.

See also A334339 for a similar conjecture.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..1000

P. Pollack and C. Pomerance, Square values of Euler's function, Bull. London Math. Soc. 46 (2014), 403-414. Alternative link.

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. (Cf. Conjecture 4.5.)

EXAMPLE

a(3) = 5 with phi(3*5) = 2^3.

a(7) = 37 with phi(7*37) = 6^3.

a(863) = 21176773 with phi(863*21176773) = 17293606056 = 2586^3.

MATHEMATICA

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];

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

tab={}; Do[m=0; Label[aa]; m=m+1; If[CQ[phi[m*n]], tab=Append[tab, m], Goto[aa]], {n, 1, 80}]; tab

PROG

(PARI) a(n) = my(m=1); while (!ispower(eulerphi(n*m), 3), m++); m; \\ Michel Marcus, Apr 23 2020

CROSSREFS

Cf. A000010, A000578, A039771, A259915, A259916, A280988, A334339.