A112374 Let T(n) = A000078(n+2), n >= 1; a(n) = smallest k such that n divides T(k).

1, 3, 6, 4, 6, 9, 8, 5, 9, 13, 20, 9, 10, 8, 6, 10, 53, 9, 48, 28, 18, 20, 35, 18, 76, 10, 9, 8, 7, 68, 20, 15, 20, 53, 30, 9, 58, 48, 78, 28, 19, 18, 63, 20, 68, 35, 28, 18, 46, 108, 76, 10, 158, 9, 52, 8, 87, 133, 18, 68, 51, 20, 46, 35, 78, 20, 17, 138, 35, 30, 230, 20, 72, 58, 76

Offset

1,2

Comments

Rank of apparition of n in the tetranacci numbers. - T. D. Noe, Dec 05 2005

This sequence is well-defined. Proof by T. D. Noe: for every prime p, Brenner proves we can find k(p) such that p divides the k(p)-th term of n-step Fibonacci. Using Brenner's methods, we know that p will also divide every j*k(p)-th term of the sequence for any j>0. We use this last fact to go to the general case: For integer m, we can find a term that m divides as follows: (1) factor m into primes: m = p1^e1 p2^e2...pr^er, (2) let K = m k(p1) k(p2)...k(pr) / (p1 p2 ... pr) (3) then m will divide the K-th term of the sequence. In general, K is much too large. However, it does show that every prime divides a term of every n-step Fibonacci sequence for n>1. - T. D. Noe, Dec 05 2005

Links

Table of n, a(n) for n=1..75.

J. L. Brenner, Linear Recurrence Relations, Amer. Math. Monthly, Vol. 61 (1954), 171-173.

T. D. Noe and J. V. Post, Primes in Fibonacci n-step and Lucas n-Step Sequences, J. Integer Seq. 8, Article 05.4.4, 2005.

Eric Weisstein's World of Mathematics, Tetranacci Number.

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.

Formula

a(n) = Min{k: n | A000078(k)}.

Mathematica

n=4; Table[a=Join[{1}, Table[0, {n-1}]]; k=0; While[k++; s=Mod[Plus@@a, i]; a=RotateLeft[a]; a[[n]]=s; s!=0]; k, {i, 100}] (* T. D. Noe, Dec 05 2005 *)

Crossrefs

Cf. A000078, A112269, A112305.

Sequence in context: A073233 A011287 A090963 * A338014 A222409 A093064

Adjacent sequences: A112371 A112372 A112373 * A112375 A112376 A112377

Keyword

easy,nonn

Author

Jonathan Vos Post, Dec 02 2005

Extensions

Corrected by T. D. Noe, Dec 05 2005

Status

approved