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A256608 Least common eventual period of a^(2^k) mod n for all a. 5
1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 2, 1, 1, 1, 2, 6, 1, 2, 4, 10, 1, 4, 2, 6, 2, 3, 1, 4, 1, 4, 1, 2, 2, 6, 6, 2, 1, 4, 2, 6, 4, 2, 10, 11, 1, 6, 4, 1, 2, 12, 6, 4, 2, 6, 3, 28, 1, 4, 4, 2, 1, 2, 4, 10, 1, 10, 2, 12, 2, 6, 6, 4, 6, 4, 2, 12, 1, 18, 4, 20, 2, 1, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

a(n) is a divisor of phi(phi(n)) (A010554).

LINKS

Ivan Neretin, Table of n, a(n) for n = 1..10000

Haifeng Xu, The largest cycles consist by the quadratic residues and Fermat primes, arXiv:1601.06509 [math.NT], 2016.

FORMULA

a(n) = A007733(A002322(n)).

a(prime(n)) = A037178(n). - Michel Marcus, Jan 27 2016

EXAMPLE

In other words, eventual period of {0..n-1} under the map x -> x^2 mod n.

For example, with n=10 the said map acts as follows. Read down the columns: the column headed 2 for example means that (repeatedly squaring mod 10), 2 goes to 4 goes to 16 = 6 (mod 10) goes to 36 = 6 mod 10 --- and has reached a fixed point.

0 1 2 3 4 5 6 7 8 9

0 1 4 9 6 5 6 9 4 1

0 1 6 1 6 5 6 1 6 1

0 1 6 1 6 5 6 1 6 1

and thus every number reaches a fixed point. This means the eventual common period is 1, hence a(10)=1.

MATHEMATICA

a[n_] := With[{lambda = CarmichaelLambda[n]}, MultiplicativeOrder[2, lambda / (2^IntegerExponent[lambda, 2])]]; Table[a[n], {n, 1, 100}] (* Jean-Fran├žois Alcover, Jan 28 2016 *)

PROG

(PARI) rpsi(n) = lcm(znstar(n)[2]); \\ A002322

pb(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ A007733

a(n) = pb(rpsi(n)); \\ Michel Marcus, Jan 28 2016

CROSSREFS

Cf. A001146, A002322, A002326, A007733, A010554, A037178.

First differs from A256607 at n=43.

LCM of entries in row n of A279185.

Sequence in context: A333570 A280726 A256607 * A279186 A164799 A274451

Adjacent sequences:  A256605 A256606 A256607 * A256609 A256610 A256611

KEYWORD

nonn

AUTHOR

Ivan Neretin, Apr 04 2015

STATUS

approved

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Last modified July 24 23:25 EDT 2021. Contains 346273 sequences. (Running on oeis4.)