A306528 Numbers k such that gcd(k, phi(k)) = gcd(k, psi(k)).

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 42, 43, 44, 46, 47, 49, 50, 52, 53, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 92, 94, 97, 98, 100

Offset: 1

Author

Torlach Rush, Feb 21 2019

Keywords

• nonn

Comments

Here phi(n) is Euler's totient function A000010 and psi(n) is Dedekind's psi function A001615.

This sequence contains all prime powers p^k where phi(p^k) and psi(p^k) are equidistant from p^k, and gcd(p^k, phi(p^k)) = gcd(p^k, psi(p^k)) = p^(k - 1). For the prime numbers themselves this is trivial since phi(p) and psi(p) differ from p by 1 and 1^0 = 1.

If prime p|k, then p*k is in the sequence if and only if k is in the sequence. - Robert Israel, Mar 05 2019

Examples

			1 is a term because gcd(1, 1) = gcd(1, 1) = 1.
2 is a term because gcd(2, 1) = gcd(2, 3) = 1.
3 is a term because gcd(3, 2) = gcd(3, 4) = 1.
4 is a term because gcd(4, 2) = gcd(4, 6) = 2.
5 is a term because gcd(5, 4) = gcd(5, 6) = 1.
6 is not a term because gcd(6, 2) <> gcd(6, 12).
7 is a term because gcd(7, 6) = gcd(7, 8) = 1.
		

Crossrefs

Cf. A000010, A001615, A009195, A306695.

Programs

Maple

filter:= proc(n) local p,F;
  F:= numtheory:-factorset(n);
  igcd(n, n*mul(1-1/p, p=F)) = igcd(n, n*mul(1+1/p,p=F))
end proc:
select(filter, [$1..200]); # Robert Israel, Mar 05 2019

PARI

dpsi(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
isok(k) = gcd(k, eulerphi(k)) == gcd(k, dpsi(k)); \\ Michel Marcus, Feb 27 2019