A029940 a(n) = Product_{d|n} phi(d).

1, 1, 2, 2, 4, 4, 6, 8, 12, 16, 10, 32, 12, 36, 64, 64, 16, 144, 18, 256, 144, 100, 22, 1024, 80, 144, 216, 864, 28, 4096, 30, 1024, 400, 256, 576, 13824, 36, 324, 576, 16384, 40, 20736, 42, 4000, 9216, 484, 46, 131072, 252, 6400, 1024, 6912, 52, 46656, 1600

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• J. R. Collis, On products of multiplicative functions, The Mathematical Gazette, Vol. 97, No. 539 (2013), pp. 263-264.

Crossrefs

Cf. A000005, A000010.

Programs

Maple

seq(mul(numtheory:-phi(i),i=numtheory:-divisors(n)), n=1..100); # Robert Israel, Nov 21 2014

Mathematica

Table[Product[EulerPhi[i], {i, Divisors[n]}], {n, 100}] (* Carl Najafi, Sep 06 2011 *)

PARI

a(n) = my(d = divisors(n)); prod(k=1, #d, eulerphi(d[k])); \\ Michel Marcus, Nov 21 2014

Formula

If n = Product_{i} p_i^e_i then a(n) = (sqrt(n) * Product_{i} (1 - 1/p_i)^(e_i/(e_i + 1))) ^ d(n), where d(n) is the number of divisors of n (Collis, 2013). - Amiram Eldar, Jun 16 2020

a(n) = Product_{k=1..n} phi(gcd(n,k))^(1/phi(n/gcd(n,k))) = Product_{k=1..n} phi(n/gcd(n,k))^(1/phi(n/gcd(n,k))). - Richard L. Ollerton, Nov 07 2021

Extensions

More terms from Carl Najafi, Sep 06 2011