A059908 - OEIS

A059908

a(n) = |{m : multiplicative order of n mod m = 3}|.

0, 1, 2, 4, 3, 2, 8, 2, 12, 5, 12, 2, 12, 2, 4, 20, 5, 6, 10, 2, 6, 14, 12, 2, 40, 9, 4, 6, 18, 10, 16, 6, 6, 8, 12, 12, 39, 2, 12, 8, 8, 6, 16, 6, 18, 26, 12, 6, 50, 3, 18, 8, 18, 2, 32, 12, 8, 20, 4, 6, 60, 2, 12, 26, 21, 4, 64, 10, 6, 8, 8, 6, 20, 14, 4, 12, 6, 4, 64, 2, 70, 7, 12, 6, 24

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OFFSET

1,3

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).

LINKS

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

FORMULA

a(n) = tau(n^3-1)-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.

EXAMPLE

a(2) = |{7}| = 1, a(3) = |{13,26}| = 2, a(4) = |{7,9,21,63}| = 4, a(5) = |{31,62,124}| = 3, a(6) = |{43,215}| = 2, a(7) = |{9,18,19,38,57,114,171,342}| = 8,...

MATHEMATICA

Table[DivisorSigma[0, n^3-1]-DivisorSigma[0, n-1], {n, 90}] (* Harvey P. Dale, Feb 03 2015 *)

CROSSREFS

Cf. A059907, A059909-A059911, A059499, A059885-A059892, A002326, A053446-A053452, A002329, A055205, A048691, A048785.

Row n=3 of A212957. - Alois P. Heinz, Oct 24 2012

Sequence in context: A283273 A269599 A368519 * A084936 A365052 A308744

Adjacent sequences: A059905 A059906 A059907 * A059909 A059910 A059911

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Feb 08 2001

STATUS

approved