A372305 a(n) = Product_{k=2..n-1} MultiplicativeOrder(k,n) where gcd(k,n)=1.

1, 1, 2, 2, 32, 2, 648, 8, 648, 32, 12500000, 8, 214990848, 648, 2048, 2048, 562949953421312, 648, 11712917736940032, 2048, 3359232, 12500000, 1377791989621882898843648, 128, 5120000000000000000, 214990848, 11712917736940032

Offset

1,3

Comments

All terms are even for n>=3.

Links

Table of n, a(n) for n=1..27.

Formula

From Ridouane Oudra, May 23 2026: (Start)

a(n) = Product_{d|lambda(n)} d^A252911(n,d), where lambda = A002322.

If n is in A033948 then:

a(n) = Product_{d|lambda(n)} d^phi(d), or also: a(n) = Product_{d|phi(n)} d^phi(d).

In particular, a(p) = Product_{d|(p-1)} d^phi(d), for p prime. (End)

Mathematica

Table[Times @@ Map[MultiplicativeOrder[#, n] &, Select[Range[2, n - 1], CoprimeQ[n, #] &]], {n, 2, 27}] (* Michael De Vlieger, Apr 25 2024 *)

Prog

(Python)
from sympy import n_order, gcd, prod
a = lambda n: prod(n_order(k, n) for k in range(2, n) if gcd(k, n)==1)
print([a(n) for n in range(1, 28)])
(PARI) a(n) = prod(k=2, n-1, if (gcd(k, n)==1, znorder(Mod(k, n)), 1)); \\ Michel Marcus, Apr 26 2024

Crossrefs

Cf. A000079, A036391, A249435, A033948, A056916, A000010, A252911.

Row products of triangle A216327.

Sequence in context: A267726 A371562 A309191 * A353911 A303568 A353928

Adjacent sequences: A372302 A372303 A372304 * A372306 A372307 A372308

Keyword

nonn

Author

Darío Clavijo, Apr 25 2024

Status

approved