cp's OEIS Frontend

A287800 Numbers n such that phi(n) * tau(n) divides n^2, but neither tau(n) nor phi(n) divides n.

%I A287800 #29 Sep 08 2022 08:46:19
%S A287800 900,2400,3840,6480,7200,11520,13056,39168,42240,79200,83232,96000,
%T A287800 126720,145200,153600,157440,174240,195840,207360,288000,300000,
%U A287800 317520,326592,387840,435600,460800,472320,480000,900000,971520,1056000,1161600,1163520,1228800,1440000
%N A287800 Numbers n such that phi(n) * tau(n) divides n^2, but neither tau(n) nor phi(n) divides n.
%C A287800 The GCD of the first 43 terms is 12. The GCD of the first 166 terms is 4. The GCD of a(2) through a(166) is 16. - _David A. Corneth_, Jun 01 2017
%H A287800 Amiram Eldar, <a href="/A287800/b287800.txt">Table of n, a(n) for n = 1..780</a> (terms below 10^10)
%e A287800 For n = 900, tau(900) = 27, phi(900) = 240 and 900^2/(27 * 240) = 125, but 900/27 = 33.33333 and 900/240 = 3.75.
%p A287800 for n from 1 to 100000 do p(n):=n^2/(tau(n)*phi(n));
%p A287800 if p(n)=floor(p(n)) and n/tau(n)<>floor(n/tau(n)) and n/phi(n)<>floor(n/phi(n)) then print(n,p(n),phi(n),tau(n)) else fi; od:
%t A287800 Select[Range[10^6], Function[n, And[Divisible[n^2, #1 #2], NoneTrue[{#1, #2}, Divisible[n, #] &]] & @@ {DivisorSigma[0, n], EulerPhi[n]}]] (* _Michael De Vlieger_, Jun 01 2017 *)
%o A287800 (PARI) is(n) = n^2 % (numdiv(n)*eulerphi(n)) == 0 && n % numdiv(n) != 0 && n % eulerphi(n) % n!=0 \\ _David A. Corneth_, Jun 01 2017
%o A287800 (Magma) [k:k in [1..1500000]| k^2 mod (EulerPhi(k) *NumberOfDivisors(k)) eq 0 and (k mod  EulerPhi(k) ne 0) and (k mod NumberOfDivisors(k) ne 0)]; // _Marius A. Burtea_, Dec 30 2019
%Y A287800 Cf. A000005, A000010, A007694, A033950, A235353.
%K A287800 nonn
%O A287800 1,1
%A A287800 _Bernard Schott_, Jun 01 2017