

A264105


a(n) = smallest k such that n divides Sum_{i=1..k} Fibonacci(i).


1



1, 2, 5, 3, 6, 5, 4, 9, 8, 6, 7, 5, 10, 15, 37, 21, 14, 8, 15, 6, 13, 9, 20, 21, 46, 27, 8, 15, 11, 39, 27, 45, 7, 14, 20, 21, 34, 15, 53, 57, 16, 15, 40, 9, 40, 20, 12, 21, 52, 99, 69, 27, 50, 8, 17, 36, 15, 11, 55, 39, 26, 27, 16, 93, 66, 29, 64, 33, 45, 20
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OFFSET

1,2


COMMENTS

Or smallest k such that n  A000071(k+2),
or smallest k such that n  A000045(k+2)1,
where Fibonacci(n) = A000045(n) and A000071(n) = A000045(n)1.
Remark: a(n) always exists for all n because the Fibonacci sequence is periodic mod n.
For all integers n, there exists an integer m such that a(m) = n. For instance if m = Fibonacci(n+2)1, then a(m) = n.


LINKS

Michel Lagneau, Table of n, a(n) for n = 1..1000


EXAMPLE

a(13) = 10 because Sum_{i=1..10} Fibonacci(i) = 1+1+2+3+5+8+13+21+34+55 = 143 = 11*13 is divisible by 13. Or 13  A000071(12) => 13143.


MAPLE

fib:= gfun:rectoproc({f(0)=0, f(1)=1, f(n)=f(n1)+f(n2)}, f(n), remember):
a:= proc(n) local k; for k from 1 do if fib(k+2)1 mod n = 0 then return k fi od end proc:
seq(a(i), i=1..1000); # Robert Israel, Nov 03 2015


MATHEMATICA

Table[s=0; k=0; While[k++; s=Mod[s+Fibonacci[k], n]; s>0]; k, {n, 100}]


PROG

(PARI) a(n) = {k=1; while(k, if( (fibonacci(k+2)1) % n == 0, return(k)); k++)} \\ Altug Alkan, Nov 05 2015


CROSSREFS

Cf. A000045, A000071.
Sequence in context: A037852 A226214 A160516 * A024871 A222072 A246007
Adjacent sequences: A264102 A264103 A264104 * A264106 A264107 A264108


KEYWORD

nonn


AUTHOR

Michel Lagneau, Nov 03 2015


STATUS

approved



