cp's OEIS Frontend

A336745 Numbers m that divide the product phi(m) * sigma(m) * tau(m), where phi is the Euler totient function (A000010), sigma is the sum of divisors function (A000203) and tau is the number of divisors function (A000005).

%I A336745 #20 Aug 05 2020 19:29:32
%S A336745 1,2,6,8,9,12,18,24,28,32,36,40,54,72,80,84,96,108,117,120,128,135,
%T A336745 144,162,196,200,216,224,234,240,243,252,270,288,324,360,384,400,405,
%U A336745 448,468,486,496,512,540,576,588,600,625,640,648,672,675,720,756,768,775,810,819
%N A336745 Numbers m that divide the product phi(m) * sigma(m) * tau(m), where phi is the Euler totient function (A000010), sigma is the sum of divisors function (A000203) and tau is the number of divisors function (A000005).
%C A336745 If s and t are terms with gcd(s, t) = 1, then s*t is another term as phi, sigma and tau are multiplicative functions.
%C A336745 The only prime term is 2 because prime p must divide 2*(p-1)*(p+1) to be a term.
%H A336745 David A. Corneth, <a href="/A336745/b336745.txt">Table of n, a(n) for n = 1..10000</a>
%e A336745 For 24, phi(24) = 8, sigma(24) = 60 and tau(24) = 8, then 8*60*8 / 24 = 160, hence 24 is a term.
%p A336745 with(numtheory):
%p A336745 filter:= m -> irem(tau(m)*phi(m)*sigma(m), m) =0:
%p A336745 select(filter,[$1..850]);
%t A336745 Select[Range[1000], Divisible[Times @@ DivisorSigma[{0, 1}, #] * EulerPhi[#], #] &] (* _Amiram Eldar_, Aug 02 2020 *)
%o A336745 (PARI) isok(m) = !(eulerphi(m)*sigma(m)*numdiv(m) % m); \\ _Michel Marcus_, Aug 05 2020
%Y A336745 Cf. A000005, A000010, A000203, A062355.
%Y A336745 Subsequences: A000396 (perfect numbers), A005820 (tri-perfect), A027687 (4-perfect), A046060 (5-multiperfect), A046061 (6-multiperfect), A007691 (multiply-perfect numbers), A336715 (m divides phi(m)*tau(m)), A004171, A005010.
%K A336745 nonn
%O A336745 1,2
%A A336745 _Bernard Schott_, Aug 02 2020