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%I #30 Sep 08 2022 08:46:24
%S 3040,9440,22240,27360,28640,30080,50560,54288,60640,67040,76752,
%T 79840,99040,105440,109888,118240,137440,152960,163040,189072,200160,
%U 201440,211536,233440,234880,239840,249216,252640,256128,256464,259040,271840,278928,296320
%N The numbers k for which gcd(k, phi(k)) + gcd(k, tau(k)) = gcd(k, sigma(k)).
%C The terms of the sequence are solutions of the equation A009191(k) + A009195(k) = A009194(k). All terms are composite numbers.
%C It seems that tau(a(n)) >= 24.
%e 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.
%t 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 *)
%o (Magma) [k: k in [1..300000]| Gcd(k,NumberOfDivisors(k))+Gcd(k,EulerPhi(k)) eq Gcd(k,SumOfDivisors(k))];
%o (PARI) isok(k) = gcd(k, numdiv(k)) + gcd(k, eulerphi(k)) == gcd(k, sigma(k)); \\ _Michel Marcus_, Oct 24 2019
%Y Cf. A000005, A000010, A000203, A009191, A009194, A009195, A328651.
%K nonn
%O 1,1
%A _Marius A. Burtea_, Oct 18 2019
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