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

Author

Jianing Song, Jan 05 2019

Keywords

• nonn

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)

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.

Programs

PARI

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

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.