A336715 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).

1, 2, 8, 9, 12, 18, 32, 36, 72, 80, 96, 108, 128, 144, 243, 288, 324, 400, 448, 486, 512, 576, 625, 720, 768, 864, 972, 1152, 1200, 1250, 1344, 1620, 1944, 2000, 2025, 2048, 2304, 2500, 2560, 2592, 2916, 3136, 3600, 3888, 4032, 4050, 4608, 5000, 5103, 5625, 6144, 6561, 6912

Offset: 1

Author

Bernard Schott, Aug 01 2020

Keywords

• nonn

Comments

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

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

Examples

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

Links

• David A. Corneth, Table of n, a(n) for n = 1..12173 (terms <= 10^15)

Crossrefs

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

Subsequences: A004171, A005010.

Programs

Maple

with(numtheory):
filter:= m-> irem(phi(m)*tau(m), m)=0:
select(filter, [$1..7000])[];

Mathematica

Select[Range[7000], Divisible[DivisorSigma[0, #] * EulerPhi[#], #] &] (* Amiram Eldar, Aug 01 2020 *)

PARI

isok(m) = (eulerphi(m)*numdiv(m) % m) == 0; \\ Michel Marcus, Aug 02 2020