A308948 -id:A308948 - cp's OEIS frontend

0, 1, 2, 1, 2, 5, 1, 1, 8, 9, 10, 5, 5, 1, 11, 1, 16, 17, 18, 1, 8, 21, 1, 1, 7, 25, 26, 1, 12, 11, 1, 1, 32, 33, 29, 17, 31, 37, 14, 1, 1, 29, 42, 21, 26, 1, 1, 1, 1, 49, 16, 1, 30, 53, 21, 1, 56, 57, 58, 41, 50, 1, 8, 1, 8, 65, 66, 33, 47, 29, 1, 1, 72, 73

Offset: 1

Jianing Song, Jul 02 2019

nonn

A214028(n) is the smallest k > 0 such that n divides A000129(k).
Let M = [{2, 1}, {1, 0}], I = [{1, 0}, {0, 1}] is the 2 X 2 identity matrix, then A214028(n) is the smallest k > 0 such that M^k == r*I (mod n) for some r such that 0 <= r < n, and a(n) gives the value r.
A214027(n) is the multiplicative order of a(n) modulo n, which can only take value 1, 2 or 4.

			For n = 7, {A000129(n) mod 7 : n > 0} = {1, 2, 5, 5, 1, 0, 1, ...}, so a(7) = 1. Also, A214028(7) = 6, and M^6 mod 7 = [{1, 0}, {0, 1}], so a(7) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000129, A214027, A214028.

Similar sequences: A217036, A308948.

a[n_] := For[k = 1, True, k++, If[Divisible[Fibonacci[k, 2], n], Return[ Mod[ Fibonacci[k+1, 2], n]]]];
Array[a, 100] (* Jean-François Alcover, Jul 05 2019 *)

a(n) = my(M=[2, 1; 1, 0]); for(k=1, 4*n/3, if((Mod(M,n)^k)[2,1]==0, return(lift((Mod(M,n)^k)[1,1]))))

Also a(n) = A000129(A214028(n)-1) mod n.
a(2^e) = 1; a(p^e) = a(p)^(p^(e-1)) mod p^e for odd primes p.
For odd primes p, a(p^e) = 1 if and only if A214028(p) == 2 (mod 4); a(p^e) = p^e - 1 if and only if 4 divides A214028(p).

cp's OEIS Frontend

A308947 a(n) = A000129(A214028(n)+1) mod n.

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