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A053446 Multiplicative order of 3 mod m, where gcd(m, 3) = 1. 17
1, 1, 2, 4, 6, 2, 4, 5, 3, 6, 4, 16, 18, 4, 5, 11, 20, 3, 6, 28, 30, 8, 16, 12, 18, 18, 4, 8, 42, 10, 11, 23, 42, 20, 6, 52, 20, 6, 28, 29, 10, 30, 16, 12, 22, 16, 12, 35, 12, 18, 18, 30, 78, 4, 8, 41, 16, 42, 10, 88, 6, 22, 23, 36, 48, 42, 20, 100, 34, 6, 52, 53, 27, 20, 12, 112, 44 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Essentially the same as A050975. - R. J. Mathar, Oct 13 2008

Let k, m be any positive numbers not divisible by 3. Let k <+> m denote that of the two numbers k + m, k + 2*m which is divisible by 3. Finally, for a number t divisible by 3, let t* = t/3^s where s is the 3-adic order of t. Let u = u(n) be the n-th number which is not divisible by 3. Consider the following algorithm of the calculating a(n), similar to the algorithm in A002326: Compute successively r_1 = (1 <+> u)*, r_2 = (r_1 <+> u)*, ..., r_h = (r_(h-1) <+> u)* and finish as soon as r_h = 1. Then a(n) = s(1 <+> u) + s(r_1 <+> u) + ... + s(r_(h-1) <+> u). Note that by a similar algorithm one can compute an arbitrary multiplicative order of a mod b, where gcd(a, b) = 1. - Vladimir Shevelev, Oct 06 2017

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = multiplicative order of 3 modulo floor((3*n-1)/2) = A001651(n), for n >= 1. - Wolfdieter Lang, Sep 28 2020

EXAMPLE

From Vladimir Shevelev, Oct 06 2017: (Start)

7 is the fifth number not divisible by 3. According to the algorithm in the comment we have in the form of a "finite continued fraction"

    1 + 14

    ------ + 7

       3

    ---------- + 14

          3

    ----------------- + 7

            3^2

    ---------------------- = 1

               3^2

Summing the exponents of 3 in the denominators, we obtain a(5) = 1 + 1 + 2 + 2 = 6. (End)

MATHEMATICA

MultiplicativeOrder[3, #] & /@ Select[ Range@ 115, GCD[3, #] == 1 &] (* Robert G. Wilson v, Apr 05 2011 *)

PROG

(PARI) lista(nn) = {for (n=1, nn, if (gcd(n, 3) == 1, print1(znorder(Mod(3, n)), ", ")); ); } \\ Michel Marcus, Feb 06 2015

(Sage)

[Mod(3, n).multiplicative_order() for n in (1..115) if gcd(n, 3) == 1] # Peter Luschny, Oct 07 2017

(GAP) List(Filtered([1..130], n->Gcd(n, 3)=1), n->OrderMod(3, n)); # Muniru A Asiru, Feb 26 2019

CROSSREFS

Cf. A001651, A002326, A050975.

Sequence in context: A075242 A161489 A050975 * A133903 A278263 A236188

Adjacent sequences:  A053443 A053444 A053445 * A053447 A053448 A053449

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

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Last modified December 7 12:17 EST 2021. Contains 349581 sequences. (Running on oeis4.)