A106279 -id:A106279 - cp's OEIS frontend

A232570 Numbers k that divide tribonacci(k) (A000073(k)).

1, 8, 16, 19, 32, 47, 53, 64, 103, 112, 128, 144, 155, 163, 192, 199, 208, 221, 224, 256, 257, 269, 272, 299, 311, 368, 397, 401, 419, 421, 448, 499, 512, 587, 599, 617, 640, 683, 757, 768, 773, 784, 863, 883, 896, 907, 911, 929, 936, 991, 1021, 1024

Offset: 1

Seiichi Manyama, Jun 17 2016

nonn

Inspired by A023172 (numbers k such that k divides Fibonacci(k)).
Includes all primes p such that x^3-x^2-x-1 has 3 distinct roots in the field GF(p) (A106279). - Robert Israel, Feb 07 2018
Includes 2^k for k >= 3. - Robert Israel, Jul 26 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000073, A023172, A106279.

with(LinearAlgebra[Modular]):
T:= (n, m)-> MatrixPower(m, Mod(m, <<0|1|0>,
    <0|0|1>, <1|1|1>>, float[8]), n)[1, 3]:
a:= proc(n) option remember; local k; if n=1
      then 1 else for k from 1+a(n-1)
      while T(k$2)>0 do od; k fi
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Feb 05 2018

trib = LinearRecurrence[{1, 1, 1}, {0, 0, 1}, 2000]; Reap[Do[If[Divisible[ trib[[n+1]], n], Print[n]; Sow[n]], {n, 1, Length[trib]-1}]][[2, 1]] (* Jean-François Alcover, Mar 22 2019 *)

require 'matrix'
def power(a, n, mod)
  return Matrix.I(a.row_size) if n == 0
  m = power(a, n >> 1, mod)
  m = (m * m).map{|i| i % mod}
  return m if n & 1 == 0
  (m * a).map{|i| i % mod}
end
def f(m, n)
  ary0 = Array.new(m, 0)
  ary0[0] = 1
  v = Vector.elements(ary0)
  ary1 = [Array.new(m, 1)]
  (0..m - 2).each{|i|
    ary2 = Array.new(m, 0)
    ary2[i] = 1
    ary1 << ary2
  }
  a = Matrix[*ary1]
  mod = n
  (power(a, n, mod) * v)[m - 1]
end
def a(n)
  (1..n).select{|i| f(3, i) == 0}
end

A106302 Period of the Fibonacci 3-step sequence A000073 mod prime(n).

Original entry on oeis.org

4, 13, 31, 48, 110, 168, 96, 360, 553, 140, 331, 469, 560, 308, 46, 52, 3541, 1860, 1519, 5113, 5328, 3120, 287, 8011, 3169, 680, 51, 1272, 990, 12883, 5376, 5720, 18907, 3864, 7400, 2850, 8269, 162, 9296, 2494, 32221, 10981, 36673, 4656, 3234, 198, 5565

Offset: 1

T. D. Noe, May 02 2005, Sep 18 2008

nonn

This sequence differs from the corresponding Lucas sequence (A106294) at n=1 and 5 because these correspond to the primes 2 and 11, which are the prime factors of -44, the discriminant of the characteristic polynomial x^3-x^2-x-1. We have a(n) < prime(n) for the primes in A106279.

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A106273, A106279, A106294, A046738.

n=3; Table[p=Prime[i]; a=Join[{1},Table[0,{n-1}]]; 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,60}]

from itertools import count
from sympy import prime
def A106302(n):
    a = b = (0,)*2+(1 % (p:= prime(n)),)
    for m in count(1):
        b = b[1:] + (sum(b) % p,)
        if a == b:
            return m # Chai Wah Wu, Feb 27 2022

a(n) = A046738(prime(n)).

A033209 Primes of form x^2 + 11*y^2.

Original entry on oeis.org

11, 47, 53, 103, 163, 199, 257, 269, 311, 397, 401, 419, 421, 499, 587, 599, 617, 683, 757, 773, 863, 883, 907, 911, 929, 991, 1021, 1087, 1109, 1123, 1181, 1237, 1291, 1307, 1367, 1433, 1439, 1543, 1567, 1571, 1609, 1621, 1697, 1699, 1753, 1873, 1907, 2003

Offset: 1

N. J. A. Sloane

Primes p such that the polynomial x^3-x^2-x-1 mod p has 3 zeros. Compare A106279. - T. D. Noe, May 02 2005

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
Tim Evink, Paul Alexander Helminck, Tribonacci numbers and primes of the form p=x^2+11y^2, arXiv:1801.04605 [math.NT], 2018.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Primes in A243651.

QuadPrimes2[1, 0, 11, 10000] (* see A106856 *)

Extended by T. D. Noe, Apr 17 2012

A106276 Number of distinct zeros of x^3-x^2-x-1 mod prime(n).

Original entry on oeis.org

1, 0, 0, 1, 2, 1, 1, 1, 0, 1, 0, 0, 1, 1, 3, 3, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 3, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 3, 1, 1, 0, 0, 0, 1, 1, 3, 1, 0, 1, 0, 1, 1, 1, 0, 3, 1, 3, 1, 1, 1, 1, 1, 1, 3, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 3, 3, 1, 3, 3, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 3, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1

Offset: 1

T. D. Noe, May 02 2005

nonn

This polynomial is the characteristic polynomial of the Fibonacci and Lucas 3-step recursions, A000073 and A001644. Similar polynomials are treated in Serre's paper. The discriminant of the polynomial is -44 = -4*11. The primes p yielding 3 distinct zeros, A106279, correspond to the periods of the sequences A000073(k) mod p and A001644(k) mod p having length less than p. The Lucas 3-step sequence mod p has two additional primes p for which the period is less than p: 2 and 11, which are factors of the discriminant -44. For p=11, the Fibonacci 3-step sequence mod p has a period of p(p-1).

J.-P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see p. 433.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.

Cf. A106273 (discriminant of the polynomial x^n-x^(n-1)-...-x-1), A106293 (period of the Lucas 3-step sequences mod prime(n)), A106282 (prime moduli for which the polynomial is irreducible).

Table[p=Prime[n]; cnt=0; Do[If[Mod[x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 150}]

A106294 Period of the Lucas 3-step sequence A001644 mod prime(n).

Original entry on oeis.org

1, 13, 31, 48, 10, 168, 96, 360, 553, 140, 331, 469, 560, 308, 46, 52, 3541, 1860, 1519, 5113, 5328, 3120, 287, 8011, 3169, 680, 51, 1272, 990, 12883, 5376, 5720, 18907, 3864, 7400, 2850, 8269, 162, 9296, 2494, 32221, 10981, 36673, 4656, 3234, 198, 5565

Offset: 1

T. D. Noe, May 02 2005

nonn

This sequence differs from the corresponding Fibonacci sequence (A106302) at n=1 and 5 because these correspond to the primes 2 and 11, which are the prime factors of -44, the discriminant of the characteristic polynomial x^3-x^2-x-1. We have a(n) < prime(n) for the primes 2, 11 and A106279.

For a prime p, the period depends on the zeros of x^3-x^2-x-1 mod p. If there are 3 zeros, then the period is < p. If there are no zeros, then the period is p^2+p+1 or a simple fraction of p^2+p+1. Also note that the period can be prime, as for p=3, 5, 31, 59, 71, 89, 97, 157, 223. When the period is prime, the orbits have a simple structure. [From T. D. Noe, Sep 18 2008]

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

Cf. A106273, A106279, A106302.

n=3; 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, 60}]

a(n) = A106293(prime(n)).

A211671 Least prime p such that the polynomial x^n - x^(n-1) - ... - 1 (mod p) has n distinct zeros.

Original entry on oeis.org

2, 11, 47, 137, 691, 25621, 59233, 2424511, 2607383, 78043403, 1032758989, 80051779

Offset: 1

T. D. Noe, Apr 18 2012

This is the characteristic polynomial of the n-step Fibonacci and Lucas sequences. For composite p, the polynomial can have more than n zeros! See A211672.

			For p = 11, x^2-x-1 = (x+3)(x+7) (mod p).
For p = 47, x^3-x^2-x-1 = (x+21)(x+30)(x+42) (mod p).
For p = 137, x^4-x^3-x^2-x-1 = (x+12)(x+79)(x+85)(x+97) (mod p).

Cf. A045468 (n=2), A106279 (n=3), A106280 (n=4), A106281 (n=5).

Cf. A211672 (for composite p).

Table[poly = x^n - Sum[x^k, {k, 0, n - 1}]; k = 1; While[p = Prime[k]; cnt = 0; Do[If[Mod[poly, p] == 0, cnt++], {x, 0, p - 1}]; cnt < n, k++]; p, {n, 5}]

a(n)={my(P=x^n-sum(k=0, n-1, x^k) ); forprime(p=2, oo, if(#polrootsmod(P,p)==n, return(p) ) );} \\ Joerg Arndt, Apr 15 2013

a(8)-a(10) from Joerg Arndt, Apr 15 2013

a(11)-a(12) from Jinyuan Wang, Apr 25 2025

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A232570 Numbers k that divide tribonacci(k) (A000073(k)).

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A106302 Period of the Fibonacci 3-step sequence A000073 mod prime(n).

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A033209 Primes of form x^2 + 11*y^2.

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A106276 Number of distinct zeros of x^3-x^2-x-1 mod prime(n).

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A106294 Period of the Lucas 3-step sequence A001644 mod prime(n).

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A211671 Least prime p such that the polynomial x^n - x^(n-1) - ... - 1 (mod p) has n distinct zeros.

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