

A322907


Entry points for the 3Fibonacci numbers A006190.


10



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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, with equality holds if and only if n = 2*7^e, e >= 1.


LINKS

Jianing Song, Table of n, a(n) for n = 1..5000


FORMULA

a(m*n) = a(m)*a(n) if gcd(m, n) = 1.
For odd primes p, a(p^e) = p^(e1)*a(p) if p^2 does not divide a(p). Any counterexample would be a 3WallSunSun prime.
a(2^e) = 3 if e = 1, 6 if e = 2 and 3*2^(e2) 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).
Sequence in context: A011209 A182649 A257698 * A071018 A144559 A155114
Adjacent sequences: A322904 A322905 A322906 * A322908 A322909 A322910


KEYWORD

nonn


AUTHOR

Jianing Song, Jan 05 2019


STATUS

approved



