A051190 a(n) = Product_{k=1..n-1} gcd(k,n).

1, 1, 1, 2, 1, 12, 1, 16, 9, 80, 1, 3456, 1, 448, 2025, 2048, 1, 186624, 1, 1024000, 35721, 11264, 1, 573308928, 625, 53248, 59049, 179830784, 1, 1007769600000, 1, 67108864, 7144929, 1114112, 37515625, 160489808068608, 1, 4980736, 89813529

Offset: 1

Author

Antti Karttunen, Oct 21 1999

Keywords

• nonn
• nice

Comments

a(n) > 1 if and only if n is composite. - Charles R Greathouse IV, Jan 04 2013

Links

• T. D. Noe, Table of n, a(n) for n = 1..200
• OEIS Wiki, Generalizations of the factorial

Crossrefs

Cf. A067911, A006579, A224479, A092287.

Programs

Haskell

a051190 n = product $ map (gcd n) [1..n-1]
-- Reinhard Zumkeller, Nov 22 2011

Maple

A051190 := proc(n) local i; mul(igcd(n, i ), i = 1..(n-1)) end;

Mathematica

a[n_] := If[PrimeQ[n], 1, Times @@ (GCD[n, #]& /@ Range[n-1])]; Table[a[n], {n, 1, 39}]  (* Jean-François Alcover, Jul 18 2012 *)
Table[Times @@ GCD[n, Range[n-1]], {n, 50}] (* T. D. Noe, Apr 12 2013 *)
Table[Product[GCD[k,n],{k,n-1}],{n,50}] (* Harvey P. Dale, Jan 29 2025 *)

PARI

a(n)=my(f=factor(n)); prod(i=1, #f[,1], prod(j=1, f[i,2], f[i,1]^(n\f[i,1]^j)))/n \\ Charles R Greathouse IV, Jan 04 2013

PARI

a(n) = prod(k=1,n-1,gcd(k,n)); /* Joerg Arndt, Apr 14 2013 */

Sage

A051190 = lambda n: mul(gcd(n,i) for i in (1..n-1))
[A051190(n) for n in (1..39)] # Peter Luschny, Apr 07 2013

Sage

# A second, faster version, based on the prime factorization of a(n):
def A051190(n):
    R = 1
    if not is_prime(n) :
        for p in primes(n//2+1):
            s = 0; r = n; t = n-1
            while r > 0 :
                r = r//p; t = t//p
                s += (r-t)*(r+t-1)
            R *= p^(s/2)
    return R
[A051190(i) for i in (1..1000)]  # Peter Luschny, Apr 08 2013

Formula

a(n) = Product_{ d divides n, d < n } d^phi(n/d). - Peter Luschny, Apr 07 2013

a(n) = A067911(n) / n. - Peter Luschny, Apr 07 2013

Product_{j=1..n} Product_{k=1..j-1} gcd(j,k), n >= 1. - Daniel Forgues, Apr 11 2013

a(n) = sqrt( (1/n) * (A092287(n) / A092287(n-1)) ). - Daniel Forgues, Apr 13 2013