A059911 a(n) = |{m : multiplicative order of n mod m = 6}|.

0, 3, 10, 16, 37, 10, 42, 24, 58, 53, 164, 26, 68, 38, 32, 68, 169, 22, 222, 38, 42, 50, 328, 40, 180, 219, 108, 26, 334, 82, 460, 82, 92, 72, 220, 108, 449, 86, 128, 80, 192, 22, 336, 110, 222, 218, 540, 84, 778, 129, 150, 80, 270, 54, 328, 356, 132, 68, 348, 22

Offset

1,2

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

Table of n, a(n) for n=1..60.

Formula

a(n) = tau(n^6-1)-tau(n^3-1)-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.

Example

a(2) = |{9,21,63}| = 3, a(3) = |{7,14,28,52,56,91,104,182,364,728}| = 10, a(4) = |{13,35,39,45,65,91,105,117,195,273,315,455,585,819,1365,4095}| = 16,...

Mathematica

a[n_] := Total[{1, -1, -1, 1} * DivisorSigma[0, n^{6, 3, 2, 1} - 1]]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jan 25 2025*)

Prog

(PARI) a(n) = if(n == 1, 0, numdiv(n^6-1) - numdiv(n^3-1) - numdiv(n^2-1) + numdiv(n-1)); \\ Amiram Eldar, Jan 25 2025

Crossrefs

Cf. A059907-A059910, A059499, A059885-A059892, A002326, A053446-A053452, A002329, A055205, A048691, A048785.

Sequence in context: A083684 A141497 A366395 * A160375 A300017 A376023

Adjacent sequences: A059908 A059909 A059910 * A059912 A059913 A059914

Keyword

easy,nonn,changed

Author

Vladeta Jovovic, Feb 08 2001

Status

approved