A110601 a(n) = phi(n)*tau(n)^2, where phi is Euler's totient function and tau(n) is the number of divisors of n.

1, 4, 8, 18, 16, 32, 24, 64, 54, 64, 40, 144, 48, 96, 128, 200, 64, 216, 72, 288, 192, 160, 88, 512, 180, 192, 288, 432, 112, 512, 120, 576, 320, 256, 384, 972, 144, 288, 384, 1024, 160, 768, 168, 720, 864, 352, 184, 1600, 378, 720, 512, 864, 208, 1152, 640

Offset: 1

Author

Emeric Deutsch, Jul 29 2005

Keywords

• nonn
• mult

Examples

			a(4)=18 because phi(4)=2 and tau(4)=3.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..2000
• Stefan Porubsky and M. G. Greening, Problem E2351, Amer. Math. Monthly, Vol. 80, No. 4 (1973), p. 436.

Crossrefs

Cf. A000005, A000010, A035116, A062355.

Programs

Magma

[EulerPhi(n)*NumberOfDivisors(n)^2: n in [1..60]]; // Vincenzo Librandi, Jun 21 2017

Maple

with(numtheory): a:=n->phi(n)*tau(n)^2: seq(a(n),n=1..70);

Mathematica

Table[EulerPhi[n]DivisorSigma[0,n]^2,{n,60}] (* Harvey P. Dale, Nov 29 2011 *)

PARI

a(n) = eulerphi(n)*numdiv(n)^2; \\ Michel Marcus, Jun 21 2017

Formula

a(n) = A000010(n) * A035116(n) = A062355(n) * A000005(n). - R. J. Mathar, Jul 26 2022

Multiplicative with a(p^e) = (e+1)^2*(p-1)*p^(e-1). - Amiram Eldar, Dec 29 2022