A106299 Primes that do not divide any term of the Lucas 3-step sequence A001644.

2, 103, 199, 211, 421, 757, 883, 907, 991, 1021, 1123, 1237, 1543, 1567, 1621, 1699, 1753, 1873, 2113, 2539, 2731, 2797, 2803, 3391, 3433, 3463, 3499, 3613, 3631, 3793, 3853, 3919, 4093, 4591, 4723, 4933, 4951, 4987, 5107, 5179, 5527, 5791, 5839, 6073

Offset

1,1

Comments

If a prime p divides a term a(k) of this sequence, then k must be less than the period of the sequence mod p. Hence these primes are found by computing A001644(k) mod p for increasing k and stopping when either A001644(k) mod p = 0 or the end of the period is reached. Interestingly, for all of these primes except 211, the period of the sequence A001644(k) mod p is (p-1)/d, where d is a small integer. The only other exceptional primes less than 1000000 are 23977 and 47093.

Links

Table of n, a(n) for n=1..44.

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

Mathematica

n=3; lst={}; Table[p=Prime[i]; a=Join[Table[ -1, {n-1}], {n}]; a=Mod[a, p]; a0=a; While[s=Mod[Plus@@a, p]; a=RotateLeft[a]; a[[n]]=s; !(a==a0 || s==0)]; If[s>0, AppendTo[lst, p]], {i, 1000}]; lst

Crossrefs

Cf. A053028 (primes not dividing any Lucas number), A106300 (primes not dividing any Lucas 4-step number), A106301 (primes not dividing any Lucas 5-step number).

Sequence in context: A055693 A357767 A215964 * A176933 A246775 A281053

Adjacent sequences: A106296 A106297 A106298 * A106300 A106301 A106302

Keyword

nonn

Author

T. D. Noe, May 02 2005

Status

approved