A318909 - OEIS

A318909

a(n) = Product_{1<=x<=n, n|(x^2-1)} x.

1, 1, 2, 3, 4, 5, 6, 105, 8, 9, 10, 385, 12, 13, 616, 945, 16, 17, 18, 1881, 2080, 21, 22, 37182145, 24, 25, 26, 5265, 28, 6061, 30, 7905, 7360, 33, 5916, 11305, 36, 37, 13300, 1384944561, 40, 15457, 42, 20769, 21736, 45, 46, 4087504225, 48, 49, 28000, 34425

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

a(n) is the product of self-inverse elements in (Z/nZ)*, where (Z/nZ)* is the multiplicative group of integers modulo n.

For n >= 2, a(n) = n - 1 iff n is in A033948. For other n, a(n) == 1 (mod n). This can also be written as: a(n) == (-1)^A034380(n) == (-1)^(A060594(n)/2) (mod n) for n >= 3.

More generally, let P(k,n) = Product_{1<=x<=n, n|(x^k-1)} x, then P(k,n) == 1 (mod n) if k is odd or n is not in A033948, P(k,n) == -1 (mod n) otherwise. Equivalently, if A046072(n) > 1 then P(k,n) == 1 (mod n), otherwise P(k,n) == (-1)^((k+1)/2) (mod n).

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

EXAMPLE

For n = 8, 1^2 == 3^2 == 5^2 == 7^2 == 1 (mod 8) so a(8) = 1*3*5*7 = 105.

For n = 12, 1^2 == 5^2 == 7^2 == 11^2 == 1 (mod 12) so a(12) = 1*5*7*11 = 385.

MAPLE

f:= proc(n) convert(map(t -> rhs(op(t)), [msolve(x^2=1, n)]), `*`) end proc:

f(1):= 1:

map(f, [$1..100]); # Robert Israel, Nov 05 2019

PROG

(PARI) a(n) = prod(i=1, n, i^(Mod(i^2-1, n)==0))

CROSSREFS