A175732 a(n) = gcd(phi(n), psi(n)).

1, 1, 2, 2, 2, 2, 2, 4, 6, 2, 2, 4, 2, 6, 8, 8, 2, 6, 2, 4, 4, 2, 2, 8, 10, 6, 18, 12, 2, 8, 2, 16, 4, 2, 24, 12, 2, 6, 8, 8, 2, 12, 2, 4, 24, 2, 2, 16, 14, 10, 8, 12, 2, 18, 8, 24, 4, 2, 2, 16, 2, 6, 12, 32, 12, 4, 2, 4, 4, 24, 2, 24, 2, 6, 40, 12, 12, 24, 2, 16, 54, 2, 2, 24, 4, 6, 8, 8, 2, 24, 8

Offset: 1

Author

Enrique Pérez Herrero, Aug 24 2010

Keywords

• nonn

Comments

a(n)^2 divides J_2(n), where J_2 is A007434.

If p > 2 is a prime, a(n) = 2. - Enrique Pérez Herrero, Jan 02 2012

Examples

			a(56) = gcd(phi(56), psi(56)) = gcd(24, 96) = 24.
		

Links

• Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A001615, A002110, A007434, A025192, A078558.

Programs

Mathematica

JordanTotient[n_, k_:1] := DivisorSum[n, #^k * MoebiusMu[n/#] &] /; (n > 0) && IntegerQ[n]; DedekindPsi[n_] := JordanTotient[n, 2]/EulerPhi[n]; A175732[n_] := GCD[EulerPhi[n], DedekindPsi[n]]; Array[A175732, 100]
f1[p_, e_] := (p - 1)*p^(e - 1); f2[p_, e_] := (p + 1)*p^(e - 1); a[1] = 1; a[n_] := Module[{f = FactorInteger[n]}, GCD[Times @@ f1 @@@ f, Times @@ f2 @@@ f]]; Array[a, 40] (* Amiram Eldar, Feb 20 2023 *)

PARI

a(n) = {my(f = factor(n)); gcd(prod(i = 1, #f~, (f[i, 1]-1)*f[i, 1]^(f[i, 2]-1)), prod(i = 1, #f~, (f[i, 1]+1)*f[i, 1]^(f[i, 2]-1)));} \\ Amiram Eldar, Feb 20 2023

Formula

a(n) = gcd(A000010(n), A001615(n)).

a(n) >= (n*2^(omega(n)-1))/rad(n).

a(A002110(n)) = A078558(n). - Enrique Pérez Herrero, Dec 04 2012

For k>=1, a(2^k) = 2^(k-1) and a(p^k) = 2*p^(k-1) if p is an odd prime. - Amiram Eldar, Feb 20 2023

a(3^n) = A025192(n). - Enrique Pérez Herrero, Jun 05 2023