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).

1, 2, 6, 8, 9, 12, 18, 24, 28, 32, 36, 40, 54, 72, 80, 84, 96, 108, 117, 120, 128, 135, 144, 162, 196, 200, 216, 224, 234, 240, 243, 252, 270, 288, 324, 360, 384, 400, 405, 448, 468, 486, 496, 512, 540, 576, 588, 600, 625, 640, 648, 672, 675, 720, 756, 768, 775, 810, 819

Offset: 1

Author

Bernard Schott, Aug 02 2020

Keywords

• nonn

Comments

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.

The only prime term is 2 because prime p must divide 2*(p-1)*(p+1) to be a term.

Examples

			For 24, phi(24) = 8, sigma(24) = 60 and tau(24) = 8, then 8*60*8 / 24 = 160, hence 24 is a term.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000010, A000203, A062355.

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.

Programs

Maple

with(numtheory):
filter:= m -> irem(tau(m)*phi(m)*sigma(m), m) =0:
select(filter,[$1..850]);

Mathematica

Select[Range[1000], Divisible[Times @@ DivisorSigma[{0, 1}, #] * EulerPhi[#], #] &] (* Amiram Eldar, Aug 02 2020 *)

PARI

isok(m) = !(eulerphi(m)*sigma(m)*numdiv(m) % m); \\ Michel Marcus, Aug 05 2020