A072523 Sum of remainders when n-th Fibonacci number is divided by all smaller Fibonacci numbers > 1.

0, 0, 0, 1, 3, 5, 10, 15, 28, 46, 79, 120, 207, 330, 540, 867, 1428, 2293, 3737, 6009, 9778, 15808, 25630, 41370, 67092, 108483, 175649, 284022, 459938, 743945, 1204113, 1947712, 3152386, 5100237, 8253262, 13352465, 21607324, 34959920

Offset

1,5

Links

Giovanni Resta, Table of n, a(n) for n = 1..1000

Formula

Conjecture: lim n->inf F(n)/a(n) = sqrt(5)/2 where F(n) is the n-th Fibonacci number and therefore lim n->inf a(n)/a(n-1) = Phi (i.e. (sqrt(5)+1)/2 or lim n->inf F(n)/F(n-1)) - Gerald McGarvey, Jul 14 2004

Example

The eighth Fibonacci number is 21; division by 2, 3, 5, 8,13 gives the remainders 1, 0, 1, 5, 8, so a(8) = 1 + 0 + 1+ 5 + 8 = 15.

Mathematica

Table[Total[Mod[Fibonacci[n], Fibonacci[Range[n-1]]]], {n, 40}] (* Harvey P. Dale, Mar 18 2015 *)

Prog

(PARI) for(n=1, 38, s=0; for(j=3, n-1, s=s+fibonacci(n)%fibonacci(j)); print1(s, ", "))

Crossrefs

Sequence in context: A326597 A008337 A077285 * A054473 A265508 A327042

Adjacent sequences: A072520 A072521 A072522 * A072524 A072525 A072526

Keyword

nonn

Author

Amarnath Murthy, Jul 31 2002

Extensions

Edited and extended by Klaus Brockhaus, Aug 02 2002

Status

approved