A106301 Primes that do not divide any term of the Lucas 5-step sequence A074048.

2, 691, 3163, 4259, 5419, 6637, 6733, 14923, 25111, 27947, 29339, 34123, 34421, 34757, 42859, 55207, 57529, 59693, 61643, 68897, 70249, 75991, 82763, 83177, 85607, 86441, 87103, 93169, 93283, 98573, 106121, 106433, 114847, 129589, 132313

Offset: 1

Author

T. D. Noe, May 02 2005

Keywords

• nonn

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 A074048(k) mod p for increasing k and stopping when either A074048(k) mod p = 0 or the end of the period is reached. Interestingly, for all of these primes, the period of the sequence A074048(k) mod p appears to be (p-1)/d, where d is a small integer.

Links

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

Crossrefs

Cf. A053028 (primes not dividing any Lucas number), A106299 (primes not dividing any Lucas 3-step number), A106300 (primes not dividing any Lucas 4-step number).

Programs

Mathematica

n=5; 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, 10000}]; lst