A067911 Product of gcd(k,n) for 1 <= k <= n.

1, 2, 3, 8, 5, 72, 7, 128, 81, 800, 11, 41472, 13, 6272, 30375, 32768, 17, 3359232, 19, 20480000, 750141, 247808, 23, 13759414272, 15625, 1384448, 1594323, 5035261952, 29, 30233088000000, 31, 2147483648, 235782657, 37879808

Offset: 1

Author

Sharon Sela (sharonsela(AT)hotmail.com), Mar 10 2002

Keywords

• nonn

Links

• T. D. Noe, Table of n, a(n) for n = 1..500
• Johann Cigler, Some remarks on the power product expansion of the q-exponential series, arXiv:2006.06242 [math.CO], 2020.
• Gottfried Helms, A dream of a (number-) sequence, 2007-2009.
• L. Toth, A survey of gcd-sum functions, J. Int. Seq. 13 (2010) # 10.8.1.

Crossrefs

Cf. A051190, A051696.

In A018804 the product is replaced by sum.

Product of terms in n-th row of A050873.

Cf. A000010 (comments on product formulas).

Programs

Maple

with(numtheory): a := n -> mul(d^phi(n/d), d = divisors(n)):
seq(a(i), i = 1..34); # Peter Luschny, Apr 07 2013

Mathematica

a[n_] := Product[d^EulerPhi[n/d], {d, Divisors[n]}];
Array[a, 34] (* Jean-François Alcover, Jun 03 2019 *)

PARI

a(n) = prod(k=1, n, gcd(k, n)); \\ Michel Marcus, Aug 23 2016

Sage

A067911 = lambda n: mul(gcd(n,i) for i in range(n))
[A067911(n) for n in (1..34)] # Peter Luschny, Apr 07 2013

Formula

a(n) = Product_{d|n} d^phi(n/d). - Vladeta Jovovic, Mar 08 2004

a(n) = n*A051190(n). - Peter Luschny, Apr 07 2013

a(n) = Product_{k=1..n} (n/gcd(n,k))^(phi(gcd(n,k))/phi(n/gcd(n,k))) where phi = A000010. - Richard L. Ollerton, Nov 07 2021

Extensions

Extended and edited by John W. Layman, Mar 14 2002