A114076 Numbers k such that k * phi(k) is a cube.

1, 4, 32, 50, 72, 225, 256, 400, 576, 900, 1944, 2048, 2166, 2312, 2646, 3200, 4107, 4563, 4608, 5202, 6075, 6250, 7200, 7225, 15125, 15552, 16384, 16428, 17328, 18252, 18496, 21168, 23762, 24300, 25600, 28125, 28900, 35378, 36864, 41616, 50000, 52488, 57600

Offset: 1

Author

Giovanni Resta, Feb 13 2006

Keywords

• nonn

Comments

From Robert Israel, Sep 06 2020: (Start)

If n > 1 is in the sequence, A071178(n) == 2 (mod 3).

If p=2^(2^k)+1 is in A019434, includes 2^a*p^b where a == 2^k-1 (mod 3) and b == 2 (mod 3).

If members m and n are coprime, then m*n is in the sequence.

If n is in the sequence and prime p divides n, then p^3*n is in the sequence. (End)

To look for terms it suffices to see if cubes have a divisors pair (k, m) such that phi(m) = k. - David A. Corneth, May 21 2024

Examples

			phi(1944) * 1944 = 1259712 = 108^3.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..8565 (first 300 terms from Robert Israel, terms <= 5*10^11)
• David A. Corneth, PARI program

Crossrefs

Aside from the first term, a subsequence of A070003. A013731 is a subsequence.

Cf. A000010, A071178, A019434.

Programs

Maple

filter:= proc(n) local F;
  F:= ifactors(n*numtheory:-phi(n))[2];
  type(map(t -> t[2]/3, F), list(integer));
end proc:
select(filter, [$1..10^5]); # Robert Israel, Sep 06 2020

Mathematica

Select[Range[57600],IntegerQ[(# EulerPhi[#])^(1/3)]&] (* Stefano Spezia, May 29 2024 *)

PARI

isok(n) = ispower(n*eulerphi(n), 3); \\ Michel Marcus, Jan 22 2014

PARI

upto(n)= res = List(); forfactored(i = 1, n, if(ispower(i[1] * eulerphi(i[2]), 3), listput(res, i[1]); ) ); res  \\ David A. Corneth, Dec 08 2022

PARI

\\ See Corneth link

Python

from sympy import integer_nthroot, totient as phi
def ok(k): return integer_nthroot(k * phi(k), 3)[1]
print([k for k in range(1, 60000) if ok(k)]) # Michael S. Branicky, Dec 08 2022

Extensions

More terms from Michel Marcus, Jan 22 2014