A123475 Product of the primitive roots of prime(n).

1, 2, 6, 15, 672, 924, 11642400, 163800, 109681110000, 5590307923200, 970377408, 134088514560000, 138960660963091968000, 874927557504000, 3456156426256013065185600000000, 30688148115024695887527936000000

Offset: 1

Author

T. D. Noe, Sep 27 2006

Keywords

• nonn

Comments

Except for n=2, we have a(n)=1 (mod prime(n)).

Examples

			a(5)=672 because the primitive roots of 11 are {2,6,7,8}.
		

References

• C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 52.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..145

Crossrefs

Cf. A060749 (primitive roots of prime(n)), A088144 (sum of primitive roots of prime(n)).

Programs

Mathematica

PrimRoots[p_] := Select[Range[p-1], MultiplicativeOrder[ #,p]==p-1&]; Table[Times@@PrimRoots[Prime[n]], {n,20}]
Times@@@Table[PrimitiveRootList[Prime[n]], {n, 20}] (* Harlan J. Brothers, Sep 02 2023 *)

PARI

vecprod(v)=prod(i=1,#v,v[i])
a(n,p=prime(n))=vecprod(select(n->znorder(Mod(n,p))==p-1,[2..p-1]))
apply(p->a(0,p), primes(20)) \\ Charles R Greathouse IV, May 15 2015

Perl

use ntheory ":all"; sub list { my $n=shift; grep { znorder($,$n) == $n-1 } 2..$n-1; } say vecprod(list($)) for @{primes(nth_prime(20))}; # Dana Jacobsen, May 15 2015