A336715 - OEIS

%I #21 Aug 05 2020 10:19:34

%S 1,2,8,9,12,18,32,36,72,80,96,108,128,144,243,288,324,400,448,486,512,

%T 576,625,720,768,864,972,1152,1200,1250,1344,1620,1944,2000,2025,2048,

%U 2304,2500,2560,2592,2916,3136,3600,3888,4032,4050,4608,5000,5103,5625,6144,6561,6912

%N Numbers m that divide the product phi(m) * tau(m), where tau is the number of divisors function (A000005) and phi is the Euler totient function (A000010).

%C Numbers of the form q = 2^(2k+1) with k>=0 (A004171) form a subsequence because tau(q) * phi(q) / q = k + 1.

%C Numbers of the form q = 9 * 2^k with k>=0 (A005010) form another subsequence because tau(q) * phi(q) / q = k+1 (also).

%H David A. Corneth, <a href="/A336715/b336715.txt">Table of n, a(n) for n = 1..12173</a> (terms <= 10^15)

%e For 80, phi(80) = 32, tau(80) = 10 and tau(80)*phi(80)/80 = 4, hence 80 is a term.

%p with(numtheory):

%p filter:= m-> irem(phi(m)*tau(m), m)=0:

%p select(filter, [$1..7000])[];

%t Select[Range[7000], Divisible[DivisorSigma[0, #] * EulerPhi[#], #] &] (* _Amiram Eldar_, Aug 01 2020 *)

%o (PARI) isok(m) = (eulerphi(m)*numdiv(m) % m) == 0; \\ _Michel Marcus_, Aug 02 2020

%Y Cf. A000010 (phi), A000005 (tau), A062355.

%Y Subsequences: A004171, A005010.

%K nonn

%O 1,2

%A _Bernard Schott_, Aug 01 2020