A306695 a(n) = gcd(n, psi(n)).

1, 1, 1, 2, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 8, 1, 18, 1, 4, 1, 2, 1, 24, 5, 2, 9, 4, 1, 6, 1, 16, 3, 2, 1, 36, 1, 2, 1, 8, 1, 6, 1, 4, 9, 2, 1, 48, 7, 10, 3, 4, 1, 54, 1, 8, 1, 2, 1, 12, 1, 2, 3, 32, 1, 6, 1, 4, 3, 2, 1, 72, 1, 2, 15, 4, 1, 6, 1, 16, 27, 2, 1, 12, 1, 2, 3, 8, 1, 18, 7, 4, 1, 2, 5, 96, 1, 14, 9, 20

Offset: 1

Author

Torlach Rush, Mar 05 2019

Keywords

• nonn

Comments

Here psi(n) is Dedekind's psi function A001615.

a(n) <= n <= A001615(n).

a(n) = n iff a(n) * 2 = A001615(n), n > 1.

a(n) = 1 iff either n=2 or n is in A255602. - Robert Israel, Mar 12 2019

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Wikipedia, Dedekind psi function

Crossrefs

Cf. A001615, A255602.

Programs

Maple

f:= proc(n) local p; igcd(n, n*mul(1+1/p, p=numtheory:-factorset(n))) end proc:
map(f, [$1..100]); # Robert Israel, Mar 11 2019

Mathematica

psi[n_] := If[n == 1, 1, n Times @@ (1 + 1/FactorInteger[n][[All, 1]])];
a[n_] := GCD[n, psi[n]];
Array[a, 100] (* Jean-François Alcover, Jun 08 2020 *)

PARI

dpsi(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
a(n) = gcd(n, dpsi(n)); \\ Michel Marcus, Mar 05 2019

Formula

a(n) = gcd(n, A001615(n)).