A106301 -id:A106301 - cp's OEIS frontend

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

T. D. Noe, May 02 2005

nonn

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.

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

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).

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

A106300 Primes that do not divide any term of the Lucas 4-step sequence A073817.

Original entry on oeis.org

2789, 3847, 4451, 4751, 5431, 6203, 8317, 9533, 9629, 9907, 10093, 11839, 13903, 13907, 14207, 15823, 16319, 16759, 19543, 20939, 21379, 21859, 25303, 26683, 29483, 30871, 31267, 31699, 32003, 32771, 33967, 34963, 36229, 37061, 39983

Offset: 1

T. D. Noe, May 02 2005

nonn

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 A073817(k) mod p for increasing k and stopping when either A073817(k) mod p = 0 or the end of the period is reached. Interestingly, for all of these primes, the period of the sequence A073817(k) mod p appears to be (p-1)/d, where d is a small integer.

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

Cf. A053028 (primes not dividing any Lucas number), A106299 (primes not dividing any Lucas 3-step number), A106301 (primes not dividing any Lucas 5-step number).

n=4; 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

cp's OEIS Frontend

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

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