A053449 -id:A053449 - cp's OEIS frontend

A059888 a(n) = |{m : multiplicative order of 6 mod m=n}|.

2, 2, 2, 4, 4, 10, 2, 8, 12, 40, 6, 108, 6, 42, 40, 48, 30, 100, 6, 332, 10, 22, 30, 376, 26, 118, 48, 332, 2, 1436, 6, 448, 54, 222, 88, 7952, 62, 54, 54, 2680, 6, 698, 30, 476, 1476, 222, 14, 7632, 28, 438, 478, 1916, 14, 1872, 84, 11896, 118, 58, 14, 784452

Offset: 1

Vladeta Jovovic, Feb 06 2001

nonn

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

Also, number of primitive factors of 6^n - 1. - Max Alekseyev, May 03 2022

Max Alekseyev, Table of n, a(n) for n = 1..430

Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), A059886 (b=4), A059887 (b=5), this sequence (b=6), A059889 (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).
Column k=6 of A212957.
Cf. A000005, A008683, A053449, A057955, A274907.

with(numtheory):
a:= n-> add(mobius(n/d)*tau(6^d-1), d=divisors(n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Oct 12 2012

a[n_] := DivisorSum[n, MoebiusMu[n/#] * DivisorSigma[0, 6^#-1] &]; Array[a, 60] (* Amiram Eldar, Jan 25 2025 *)

a(n) = sumdiv(n, d, moebius(n/d) * numdiv(6^d-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = Sum_{ d divides n } mu(n/d)*tau(6^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).

A050978 Haupt-exponents of 6 modulo integers relatively prime to 6.

Original entry on oeis.org

1, 2, 10, 12, 16, 9, 11, 5, 14, 6, 2, 4, 40, 3, 23, 14, 26, 10, 58, 60, 12, 33, 35, 36, 10, 78, 82, 16, 88, 12, 9, 12, 10, 102, 106, 108, 112, 11, 16, 110, 25, 126, 130, 18, 136, 23, 60, 14, 37, 150, 6, 156, 22, 27, 83, 156, 43, 10, 178, 60, 4, 80, 19, 96, 14, 198, 14

Offset: 1

Eric W. Weisstein

nonn

Eric Weisstein's World of Mathematics, Multiplicative Order

Cf. A002326, A002329, A053449.

A054711 Multiplicative order of 11 mod n, where gcd(n, 11) = 1.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 3, 2, 6, 1, 2, 12, 3, 2, 4, 16, 6, 3, 2, 6, 22, 2, 5, 12, 18, 6, 28, 2, 30, 8, 16, 3, 6, 6, 3, 12, 2, 40, 6, 7, 6, 22, 46, 4, 21, 5, 16, 12, 26, 18, 6, 6, 28, 58, 2, 4, 30, 6, 16, 12, 66, 16, 22, 3, 70, 6, 72, 6, 10, 6, 12, 39, 4, 54, 40, 41

Offset: 1

Henry Bottomley, Apr 20 2000

The original version "Number of powers of 11 modulo n" that was similar to A054703-A054717 is now in A351524. - Georg Fischer, Feb 13 2022

Cf. A053446 (of 3 mod n), A053448 (5), A053449 (6), A053450 (7), A053452 (9).

Cf. A351524.

MultiplicativeOrder[11, #] & /@ Select[ Range@ 90, GCD[11, #] == 1 &] (* Robert G. Wilson v, Apr 05 2011 *)

lista(nn) = {for(n=1, nn, if (gcd(n, 11) == 1, print1(znorder(Mod(11, n)), ", ")););} \\ Michel Marcus, Feb 09 2015

Corrected by Michel Marcus, Feb 11 2015

cp's OEIS Frontend

A059888 a(n) = |{m : multiplicative order of 6 mod m=n}|.

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A050978 Haupt-exponents of 6 modulo integers relatively prime to 6.

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A054711 Multiplicative order of 11 mod n, where gcd(n, 11) = 1.

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