OFFSET
1,2
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
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
EXAMPLE
phi(1944) * 1944 = 1259712 = 108^3.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Giovanni Resta, Feb 13 2006
EXTENSIONS
More terms from Michel Marcus, Jan 22 2014
STATUS
approved