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

1,2

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).

Links

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

Example

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

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 *)

Prog

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

Crossrefs

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

Subsequences: A004171, A005010.

Sequence in context: A046526 A279373 A057529 * A120737 A081381 A235524

Adjacent sequences: A336712 A336713 A336714 * A336716 A336717 A336718

Keyword

nonn

Author

Bernard Schott, Aug 01 2020

Status

approved