A326416 The numbers k for which gcd(k, phi(k)) + gcd(k, tau(k)) = gcd(k, sigma(k)).

3040, 9440, 22240, 27360, 28640, 30080, 50560, 54288, 60640, 67040, 76752, 79840, 99040, 105440, 109888, 118240, 137440, 152960, 163040, 189072, 200160, 201440, 211536, 233440, 234880, 239840, 249216, 252640, 256128, 256464, 259040, 271840, 278928, 296320

Offset: 1

Author

Marius A. Burtea, Oct 18 2019

Keywords

• nonn

Comments

The terms of the sequence are solutions of the equation A009191(k) + A009195(k) = A009194(k). All terms are composite numbers.

It seems that tau(a(n)) >= 24.

Examples

			For k = 3040 = 2^5 * 5 * 19, phi(k) = 2^4 * 4 * 18 = 2^7 * 3^2, tau(k) = 6* 2* 2 = 2^3 * 3, sigma(k) = 2^3 * 3^3 * 5 *7, gcd(k,phi(k)) + gcd(k tau(k)) = 2^5 + 2^3 = 40 and gcd(k,sigma(k)) = 2^3 * 5 = 40.
		

Crossrefs

Cf. A000005, A000010, A000203, A009191, A009194, A009195, A328651.

Programs

Magma

[k: k in [1..300000]| Gcd(k,NumberOfDivisors(k))+Gcd(k,EulerPhi(k)) eq Gcd(k,SumOfDivisors(k))];

Mathematica

aQ[n_] := GCD[n, EulerPhi[n]] + GCD[n, DivisorSigma[0, n]] ==  GCD[n, DivisorSigma[1, n]]; Select[Range[300000], aQ] (* Amiram Eldar, Oct 23 2019 *)

PARI

isok(k) = gcd(k, numdiv(k)) + gcd(k, eulerphi(k)) == gcd(k, sigma(k)); \\ Michel Marcus, Oct 24 2019