A322907 Entry points for the 3-Fibonacci numbers A006190.

1, 3, 2, 6, 3, 6, 8, 6, 6, 3, 4, 6, 13, 24, 6, 12, 8, 6, 20, 6, 8, 12, 22, 6, 15, 39, 18, 24, 7, 6, 32, 24, 4, 24, 24, 6, 19, 60, 26, 6, 7, 24, 42, 12, 6, 66, 48, 12, 56, 15, 8, 78, 26, 18, 12, 24, 20, 21, 12, 6, 30, 96, 24, 48, 39, 12, 68, 24, 22, 24, 72, 6

Offset

1,2

Comments

a(n) is the smallest k > 0 such that n divides A006190(k).

a(n) is also called the rank of A006190(n) modulo n.

For primes p == 1, 9, 17, 25, 29, 49 (mod 52), a(p) divides (p - 1)/2.

For primes p == 3, 23, 27, 35, 43, 51 (mod 52), a(p) divides p - 1, but a(p) does not divide (p - 1)/2.

For primes p == 5, 21, 33, 37, 41, 45 (mod 52), a(p) divides (p + 1)/2.

For primes p == 7, 11, 15, 19, 31, 47 (mod 52), a(p) divides p + 1, but a(p) does not divide (p + 1)/2.

a(n) <= (12/7)*n for all n, where the equality holds if and only if n = 2*7^e, e >= 1.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Jianing Song)

Formula

a(m*n) = a(m)*a(n) if gcd(m, n) = 1.

For odd primes p, a(p^e) = p^(e-1)*a(p) if p^2 does not divide a(p). Any counterexample would be a 3-Wall-Sun-Sun prime.

a(2^e) = 3 if e = 1, 6 if e = 2 and 3*2^(e-2) if e >= 3. a(13^e) = 13^e, e >= 1.

Prog

(PARI) A006190(m) = ([3, 1; 1, 0]^m)[2, 1]
a(n) = my(i=1); while(A006190(i)%n!=0, i++); i

Crossrefs

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = k*x(n+1) + x(n). Then the periods, ranks and the ratios of the periods to the ranks modulo a given integer n are given by:

k = 1: A001175 (periods), A001177 (ranks), A001176 (ratios).

k = 2: A175181 (periods), A214028 (ranks), A214027 (ratios).

k = 3: A175182 (periods), this sequence (ranks), A322906 (ratios).

Cf. A006190.

Sequence in context: A182649 A257698 A364384 * A071018 A144559 A155114

Adjacent sequences: A322904 A322905 A322906 * A322908 A322909 A322910

Keyword

nonn

Author

Jianing Song, Jan 05 2019

Status

approved