A377121 Numbers whose totient is refactorable.

1, 2, 3, 4, 6, 13, 15, 16, 19, 20, 21, 24, 26, 27, 28, 30, 35, 36, 37, 38, 39, 41, 42, 45, 52, 54, 55, 56, 57, 61, 63, 70, 72, 73, 74, 75, 76, 77, 78, 82, 84, 87, 88, 89, 90, 91, 93, 95, 97, 99, 100, 108, 109, 110, 111, 114, 115, 116, 117, 119, 122, 123, 124, 126, 129, 132, 133, 135, 137, 146, 147, 148, 150, 152, 153

Offset: 1

Author

Waldemar Puszkarz, Oct 17 2024

Keywords

• nonn

Comments

If k is an odd term, 2*k is also a term due to the multiplicative nature of the totient function and their totients are equal.

Examples

			3, 4, and 6 are terms as their totients equal 2, which is the second refactorable number.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A033950.

Programs

Maple

filter:= proc(n) local p;
  p:= numtheory:-phi(n);
  p mod numtheory:-tau(p) = 0
end proc:
select(filter, [$1..200]); # Robert Israel, Dec 12 2024

Mathematica

Select[Range[200], Mod[EulerPhi[#], DivisorSigma[0,EulerPhi[#]]]==0&]

PARI

for(n=1, 200, a=eulerphi(n); a%numdiv(a)==0&&print1(n", "))

Python

from sympy import divisor_sigma, totient
for i in range(1, 200):
    if totient(i)%divisor_sigma(totient(i), 0)==0:
        print(i, end=", ")