A059907 a(n) = |{m : multiplicative order of n mod m = 2}|.

0, 1, 2, 2, 5, 2, 6, 4, 6, 3, 12, 2, 10, 6, 8, 4, 13, 2, 18, 6, 10, 4, 16, 4, 12, 9, 12, 4, 26, 2, 20, 6, 8, 12, 20, 4, 15, 6, 16, 4, 32, 2, 24, 10, 10, 6, 20, 4, 26, 9, 18, 4, 26, 6, 32, 12, 12, 4, 28, 2, 20, 10, 12, 18, 25, 4, 24, 6, 26, 4, 52, 2, 18, 10, 12, 18, 26, 4, 40, 8, 14, 5, 28

Offset: 1

Author

Vladeta Jovovic, Feb 08 2001

Keywords

• easy
• nonn

Comments

The multiplicative order of a mod m, GCD(a,m) = 1, is the smallest natural number d for which a^d = 1 (mod m).

Examples

			a(2) = |{3}| = 1, a(3) = |{4,8}| = 2, a(4) = |{5,15}| = 2, a(5) = |{3,6,8,12,24}| = 5, a(6) = |{7,35}| = 2, a(7) = |{4,8,12,16,24,48}| = 6,...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002329, A059908-A059911, A059499, A059885-A059892, A002326, A053446-A053452, A055205, A048691, A048785, A347191.

Row n=2 of A212957. - Alois P. Heinz, Oct 24 2012

Programs

Maple

with(numtheory):f := n->tau(n^2-1)-tau(n-1):for n from 1 to 100 do printf(`%d,`,f(n)) od:

Mathematica

a[n_] := Subtract @@ DivisorSigma[0, {n^2-1, n-1}]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jan 25 2025 *)

PARI

a(n) = if(n == 1, 0, numdiv(n^2-1) - numdiv(n-1)); \\ Amiram Eldar, Jan 25 2025

Formula

a(n) = tau(n^2-1)-tau(n-1), where tau(n) = number of divisors of n A000005. Generally, if b(n, r) = |{m : multiplicative order of n mod m = r}| then b(n, r) = Sum_{d|r} mu(d)*tau(n^(r/d)-1), where mu(n) = Moebius function A008683.