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A106298 Period of the Lucas 5-step sequence A074048 mod prime(n). 5
1, 104, 781, 2801, 16105, 30941, 88741, 13032, 12166, 70728, 190861, 1926221, 2896405, 79506, 736, 8042221, 102689, 3720, 20151120, 2863280, 546120, 39449441, 48030024, 3690720, 29509760, 104060400, 37516960, 132316201, 28231632, 6384, 86714880, 2248090, 3128 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This sequence is the same as the period of Fibonacci 5-step sequence (A106304) mod prime(n) except for n=1 and 109, which correspond to the primes 2 and 599 because 9584 is the discriminant of the characteristic polynomial x^5-x^4-x^3-x^2-x-1 and the prime factors of 9584 are 2 and 599. We have a(n) < prime(n) for the primes 2, 599 and A106281.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..82

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

FORMULA

a(n) = A106297(prime(n)).

MATHEMATICA

n=5; Table[p=Prime[i]; a=Join[Table[ -1, {n-1}], {n}]; a=Mod[a, p]; a0=a; k=0; While[k++; s=Mod[Plus@@a, p]; a=RotateLeft[a]; a[[n]]=s; a!=a0]; k, {i, 40}]

PROG

(Python)

from itertools import count

from sympy import prime

def A106298(n):

    a = b = (5%(p:=prime(n)), 1%p, 7%p, 3%p, 15%p)

    s = sum(b) % p

    for m in count(1):

        b, s = b[1:] + (s, ), (s+s-b[0]) % p

        if a == b:

            return m # Chai Wah Wu, Feb 22-27 2022

CROSSREFS

Cf. A106281 (primes p such that x^5-x^4-x^3-x^2-x-1 mod p has 5 distinct zeros).

Sequence in context: A250677 A206021 A206014 * A337145 A230028 A340898

Adjacent sequences:  A106295 A106296 A106297 * A106299 A106300 A106301

KEYWORD

nonn

AUTHOR

T. D. Noe, May 02 2005

EXTENSIONS

a(31)-a(33) from Chai Wah Wu, Feb 27 2022

STATUS

approved

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Last modified August 8 08:38 EDT 2022. Contains 356003 sequences. (Running on oeis4.)