A053030 -id:A053030 - cp's OEIS frontend

A001176 Number of zeros in fundamental period of Fibonacci numbers mod n.

1, 1, 2, 1, 4, 2, 2, 2, 2, 4, 1, 2, 4, 2, 2, 2, 4, 2, 1, 2, 2, 1, 2, 2, 4, 4, 2, 2, 1, 2, 1, 2, 2, 4, 2, 2, 4, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 1, 1, 2, 4, 1, 2, 2, 4, 2, 2, 2, 2, 2, 1, 2, 4, 4, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 1, 2, 2, 2, 2

Offset: 1

N. J. A. Sloane

If the Fibonacci numbers are indexed so that 3 is the fourth number, then if the modulo base is a Fibonacci number (>= 3) with an even index, the period has 2 zeros. If the base is a Fibonacci number (>= 5) with an odd index, the period has 4 zeros. - Kerry Mitchell, Dec 11 2005

For a proof that A001177(n) divides the period length A001175(n) for n >= 1, see, e.g., the Vajda reference, p. 73. This comment refers to the present first formula. - Wolfdieter Lang, Jan 19 2015

			{F(n) mod 1} has fundamental period (0) with 1 zero.
{F(n) mod 2} has fundamental period (0,1,1) with 1 zero.
{F(n) mod 3} has fundamental period (0,1,1,2,0,2,2,1) with 2 zeros.
{F(n) mod 4} has fundamental period (0,1,1,2,3,1), with 1 zero.
{F(n) mod 5} has fundamental period (0,1,1,2,3,0,3,3,1,4,0,4,4,3,2,0,2,2,4,1) with 4 zeros.

B. H. Hannon and W. L. Morris, Tables of Arithmetical Functions Related to the Fibonacci Numbers. Report ORNL-4261, Oak Ridge National Laboratory, Oak Ridge, Tennessee, Jun 1968.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.

T. D. Noe, Table of n, a(n) for n = 1..1000
Brennan Benfield and Michelle Manes, The Fibonacci Sequence is Normal Base 10, arXiv:2202.08986 [math.NT], 2022.
J. D. Fulton and W. L. Morris, On arithmetical functions related to the Fibonacci numbers, Acta Arithmetica, 16 (1969), 105-110.
B. H. Hannon and W. L. Morris, Tables of Arithmetical Functions Related to the Fibonacci Numbers [Annotated and scanned copy]
M. Renault, Fibonacci sequence modulo m
Review of B. H. Hannon and W. L. Morris tables, Math. Comp., 23 (1969), 459-460.

Cf. A001175, A001177, A053027, A053028, A053029, A053030, A053031, A053032.

Cf. A235715.

a001176 1 = 1
a001176 n = f 1 ps 0 where
   f 0 (1 : xs) z = z
   f _ (x : xs) z = f x xs (z + 0 ^ x)
   ps = 1 : 1 : zipWith (\u v -> (u + v) `mod` n) (tail ps) ps
-- Reinhard Zumkeller, Jan 15 2014

With[{fibs=Fibonacci[Range[2000]]},Table[Count[FindTransientRepeat[ Mod[ fibs, n], 3][[2]],0],{n,110}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 26 2016 *)

a(n) = A001175(n)/A001177(n) for n >= 1.
a(n) = ord(n, fibonacci(A001177(n) + 1)), where ord(n, a) is the multiplicative order of a modulo n. - Mircea Merca, Jan 03 2011
a(n) = A128924(n,1). - Reinhard Zumkeller, Jan 17 2014
From Isaac Saffold, Aug 30 2018: (Start)
With the sole exception of a(8) = 2,
  a(p^k) = 1 if A007814(A001175(p^k)) < 2.
  a(p^k) = 4 if A007814(A001175(p^k)) = 2.
  a(p^k) = 2 if A007814(A001175(p^k)) > 2. (End)
From Jianing Song, Sep 01 2018: (Start)
a(2^e) = 1 if e <= 2, otherwise 2. For odd primes p, a(p^e) = 4 if A001177(p) is odd; 1 if A001177(p) is even but not divisible by 4; 2 if A001177(p) is divisible by 4.
a(n) = 2 for n == 0, 3, 7, 8, 12, 15 (mod 20). a(p^e) = 1 if primes p == 11, 19 (mod 20); 4 if p == 13, 17 (mod 20). Conjecture: 1/6 of the primes congruent to 1 or 9 mod 40 satisfy a(p^e) = 1, 2/3 of them satisfy a(p^e) = 2 and 1/6 of them satisfy a(p^e) = 4; also, 1/2 of the primes congruent to 21 or 29 mod 40 satisfy a(p^e) = 1 and 1/2 of them satisfy a(p^e) = 4. (End)

Better description and more terms from Henry Bottomley, Feb 01 2000
Examples from David W. Wilson, Jan 05 2005
Replaced the old Renault link with a working one. - Wolfdieter Lang, Jan 17 2015

A053028 Odd primes p with 4 zeros in any period of the Fibonacci numbers mod p.

Original entry on oeis.org

5, 13, 17, 37, 53, 61, 73, 89, 97, 109, 113, 137, 149, 157, 173, 193, 197, 233, 257, 269, 277, 293, 313, 317, 337, 353, 373, 389, 397, 421, 433, 457, 557, 577, 593, 613, 617, 653, 661, 673, 677, 701, 733, 757, 761, 773, 797, 821, 829, 853, 857, 877, 937, 953

Offset: 1

Henry Bottomley, Feb 23 2000

nonn

Also, primes that do not divide any Lucas number. - T. D. Noe, Jul 25 2003
Although every prime divides some Fibonacci number, this is not true for the Lucas numbers. In fact, exactly 1/3 of all primes do not divide any Lucas number. See Lagarias and Moree for more details. The Lucas numbers separate the primes into three disjoint sets: (A053028) primes that do not divide any Lucas number, (A053027) primes that divide Lucas numbers of even index and (A053032) primes that divide Lucas numbers of odd index. - T. D. Noe, Jul 25 2003; revised by N. J. A. Sloane, Feb 21 2004
From Jianing Song, Jun 16 2024: (Start)
Primes p such that A001176(p) = 4.
For p > 2, p is in this sequence if and only if A001175(p) == 4 (mod 8), and if and only if A001177(p) is odd. For a proof of the equivalence between A001176(p) = 4 and A001177(p) being odd, see Section 2 of my link below.
This sequence contains all primes congruent to 13, 17 (mod 20). This corresponds to case (1) for k = 3 in the Conclusion of Section 1 of my link below. (End) [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]

T. D. Noe, Table of n, a(n) for n=1..1000
C. Ballot and M. Elia, Rank and period of primes in the Fibonacci sequence; a trichotomy, Fib. Quart., 45 (No. 1, 2007), 56-63 (The sequence B2).
Nicholas Bragman and Eric Rowland, Limiting density of the Fibonacci sequence modulo powers of p, arXiv:2202.00704 [math.NT], 2022.
J. C. Lagarias, The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math., 118. No. 2, (1985), 449-461.
J. C. Lagarias, Errata to: The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math., 162, No. 2, (1994), 393-396.
Diego Marques and Pavel Trojovsky, The order of appearance of the product of five consecutive Lucas numbers, Tatra Mountains Math. Publ. 59 (2014), 65-77.
Pieter Moree, Counting Divisors of Lucas Numbers, Pacific J. Math, Vol. 186, No. 2, 1998, pp. 267-284.
M. Renault, Fibonacci sequence modulo m
H. Sedaghat, Zero-Avoiding Solutions of the Fibonacci Recurrence Modulo A Prime, Fibonacci Quart. 52 (2014), no. 1, 39-45. See p. 45.
Jianing Song, Lucas sequences and entry point modulo p
Eric Weisstein's World of Mathematics, Lucas Number

Cf. A001175, A001177.
Cf. A000204 (Lucas numbers), A001602 (index of the smallest Fibonacci number divisible by prime(n)).
Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.
                             |   m=1    |   m=2   |   m=3
-----------------------------+----------+---------+---------
The sequence {x(n)}          | A000045  | A000129 | A006190
The sequence {w(k)}          | A001176  | A214027 | A322906
Primes p such that w(p) = 1  | A112860* | A309580 | A309586
Primes p such that w(p) = 2  | A053027  | A309581 | A309587
Primes p such that w(p) = 4  | this seq | A261580 | A309588
Numbers k such that w(k) = 1 | A053031  | A309583 | A309591
Numbers k such that w(k) = 2 | A053030  | A309584 | A309592
Numbers k such that w(k) = 4 | A053029  | A309585 | A309593
* and also A053032 U {2}

Lucas[n_] := Fibonacci[n+1] + Fibonacci[n-1]; badP={}; Do[p=Prime[n]; k=1; While[k0, k++ ]; If[k==p, AppendTo[badP, p]], {n, 200}]; badP

A prime p = prime(i) is in this sequence if p > 2 and A001602(i) is odd. - T. D. Noe, Jul 25 2003

Edited: Name clarified. Moree and Renault link updated. Ballot and Elia reference linked. - Wolfdieter Lang, Jan 20 2015

A053032 Odd primes p with one zero in Fibonacci numbers mod p.

Original entry on oeis.org

11, 19, 29, 31, 59, 71, 79, 101, 131, 139, 151, 179, 181, 191, 199, 211, 229, 239, 251, 271, 311, 331, 349, 359, 379, 419, 431, 439, 461, 479, 491, 499, 509, 521, 541, 571, 599, 619, 631, 659, 691, 709, 719, 739, 751, 809, 811, 839, 859, 911, 919, 941, 971

Offset: 1

Henry Bottomley, Feb 23 2000

nonn

Also, odd primes that divide Lucas numbers of odd index. - T. D. Noe, Jul 25 2003
From Charles R Greathouse IV, Dec 14 2016: (Start)
It seems that this sequence contains about 1/3 of the primes. In particular, members of this sequence constitute:
        35 of the first 10^2 primes
       330 of the first 10^3 primes
      3328 of the first 10^4 primes
     33371 of the first 10^5 primes
    333329 of the first 10^6 primes
   3333720 of the first 10^7 primes
  33333463 of the first 10^8 primes
etc. (End)
Of the Fibonacci-like sequences modulo a prime p that are not A000004, one of them has a period length less than A001175(p) if and only if p = 5 or p is in this sequence. - Isaac Saffold, Dec 18 2018
Odd primes in A053031. - Jianing Song, Jun 19 2019

			From _Michael B. Porter_, Jan 25 2019: (Start)
The Fibonacci numbers (mod 7) repeat the pattern 0, 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1. Since there are two zeros, 7 is not in the sequence.
The Fibonacci numbers (mod 11) repeat the pattern 0, 1, 1, 2, 3, 5, 8, 2, 10, 1 which has only one zero, so 11 is in the sequence.
(End)

T. D. Noe, Table of n, a(n) for n = 1..1000
C. Ballot and M. Elia, Rank and period of primes in the Fibonacci sequence; a trichotomy, Fib. Quart., 45 (No. 1, 2007), 56-63 (The sequence B1).
Nicholas Bragman and Eric Rowland, Limiting density of the Fibonacci sequence modulo powers of p, arXiv:2202.00704 [math.NT], 2022.
M. Renault, Fibonacci sequence modulo m

Cf. A001175, A001177. See A112860 for another version.
Cf. A000204 (Lucas numbers), A001602 (index of the smallest Fibonacci number divisible by prime(n)).
Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.
                             |    m=1     |   m=2   |   m=3
-----------------------------+------------+---------+---------
The sequence {x(n)}          | A000045    | A000129 | A006190
The sequence {w(k)}          | A001176    | A214027 | A322906
Primes p such that w(p) = 1  | A112860*   | A309580 | A309586
Primes p such that w(p) = 2  | A053027**  | A309581 | A309587
Primes p such that w(p) = 4  | A053028*** | A261580 | A309588
Numbers k such that w(k) = 1 | A053031    | A309583 | A309591
Numbers k such that w(k) = 2 | A053030    | A309584 | A309592
Numbers k such that w(k) = 4 | A053029    | A309585 | A309593
* and also this sequence U {2}
** also primes dividing Lucas numbers of even index
*** also primes dividing no Lucas number

Prime@ Rest@ Position[Table[Count[Drop[NestWhile[Append[#, Mod[Total@ Take[#, -2], n]] &, {1, 1}, If[Length@ # < 3, True, Take[#, -2] != {1, 1}] &], -2], 0], {n, Prime@ Range@ 168}], 1][[All, 1]] (* Michael De Vlieger, Aug 08 2018 *)

fibmod(n,m)=(Mod([1, 1; 1, 0], m)^n)[1, 2]
is(n)=my(k=n+[0, -1, 1, 1, -1][n%5+1]); k>>=valuation(k,2)-1; fibmod(k,n)==0 && fibmod(k/2,n) && isprime(n) \\ Charles R Greathouse IV, Dec 14 2016

A prime p = prime(i) is in this sequence if p > 2 and A001602(i)/2 is odd. - T. D. Noe, Jul 25 2003

A053027 Odd primes p with 2 zeros in Fibonacci numbers mod p.

Original entry on oeis.org

3, 7, 23, 41, 43, 47, 67, 83, 103, 107, 127, 163, 167, 223, 227, 241, 263, 281, 283, 307, 347, 367, 383, 401, 409, 443, 449, 463, 467, 487, 503, 523, 547, 563, 569, 587, 601, 607, 641, 643, 647, 683, 727, 743, 769, 787, 823, 827, 863, 881, 883, 887, 907, 929

Offset: 1

Henry Bottomley, Feb 23 2000

nonn

Also, odd primes that divide Lucas numbers of even index. - T. D. Noe, Jul 25 2003
Primes in A053030. - Jianing Song, Jun 19 2019
From Jianing Song, Jun 16 2024: (Start)
Primes p such that A001176(p) = 2.
For p > 2, p is in this sequence if and only if 8 divides of A001175(p), and if and only if 4 divides A001177(p). For a proof of the equivalence between A001176(p) = 2 and 4 dividing A001177(p), see Section 2 of my link below.
This sequence contains all primes congruent to 3, 7 (mod 20). This corresponds to case (2) for k = 3 in the Conclusion of Section 1 of my link below.
Conjecturely, this sequence has density 1/3 in the primes. (End) [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]

T. D. Noe, Table of n, a(n) for n=1..1000
C. Ballot and M. Elia, Rank and period of primes in the Fibonacci sequence; a trichotomy, Fib. Quart., 45 (No. 1, 2007), 56-63 (The sequence B3).
Nicholas Bragman and Eric Rowland, Limiting density of the Fibonacci sequence modulo powers of p, arXiv:2202.00704 [math.NT], 2022.
M. Renault, Fibonacci sequence modulo m
Jianing Song, Lucas sequences and entry point modulo p

Cf. A001175, A001177.
Cf. A000204 (Lucas numbers), A001602 (index of the smallest Fibonacci number divisible by prime(n)).
Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.
                             |    m=1    |   m=2   |   m=3
-----------------------------+-----------+---------+---------
The sequence {x(n)}          | A000045   | A000129 | A006190
The sequence {w(k)}          | A001176   | A214027 | A322906
Primes p such that w(p) = 1  | A112860*  | A309580 | A309586
Primes p such that w(p) = 2  | this seq  | A309581 | A309587
Primes p such that w(p) = 4  | A053028** | A261580 | A309588
Numbers k such that w(k) = 1 | A053031   | A309583 | A309591
Numbers k such that w(k) = 2 | A053030   | A309584 | A309592
Numbers k such that w(k) = 4 | A053029   | A309585 | A309593
* and also A053032 (primes dividing Lucas numbers of odd index) U {2}
** also primes dividing no Lucas number

A prime p = prime(i) is in this sequence if p > 2 and A001602(i)/2 is even. - T. D. Noe, Jul 25 2003

A053029 Numbers with 4 zeros in Fibonacci numbers mod m.

Original entry on oeis.org

5, 10, 13, 17, 25, 26, 34, 37, 50, 53, 61, 65, 73, 74, 85, 89, 97, 106, 109, 113, 122, 125, 130, 137, 146, 149, 157, 169, 170, 173, 178, 185, 193, 194, 197, 218, 221, 226, 233, 250, 257, 265, 269, 274, 277, 289, 293, 298, 305, 313, 314, 317, 325, 337, 338, 346

Offset: 1

Henry Bottomley, Feb 23 2000

nonn

Conjecture: m is on this list iff m is an odd number all of whose factors are on this list or m is twice such an odd number.

A001176(a(n)) = A128924(a(n),1) = 4. - Reinhard Zumkeller, Jan 17 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Brennan Benfield and Michelle Manes, The Fibonacci Sequence is Normal Base 10, arXiv:2202.08986 [math.NT], 2022.
Brennan Benfield and Oliver Lippard, Connecting Zeros in Pisano Periods to Prime Factors of K-Fibonacci Numbers, arXiv:2407.20048 [math.NT], 2024.
M. Renault, Fibonacci sequence modulo m

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.
                             |   m=1    |   m=2   |   m=3
-----------------------------+----------+---------+---------
The sequence {x(n)}          | A000045  | A000129 | A006190
The sequence {w(k)}          | A001176  | A214027 | A322906
Primes p such that w(p) = 1  | A112860* | A309580 | A309586
Primes p such that w(p) = 2  | A053027  | A309581 | A309587
Primes p such that w(p) = 4  | A053028  | A261580 | A309588
Numbers k such that w(k) = 1 | A053031  | A309583 | A309591
Numbers k such that w(k) = 2 | A053030  | A309584 | A309592
Numbers k such that w(k) = 4 | this seq | A309585 | A309593
* and also A053032 U {2}

a053029 n = a053029_list !! (n-1)
a053029_list = filter ((== 4) . a001176) [1..]
-- Reinhard Zumkeller, Jan 17 2014

A053031 Numbers with 1 zero in Fibonacci numbers mod m.

Original entry on oeis.org

1, 2, 4, 11, 19, 22, 29, 31, 38, 44, 58, 59, 62, 71, 76, 79, 101, 116, 118, 121, 124, 131, 139, 142, 151, 158, 179, 181, 191, 199, 202, 209, 211, 229, 236, 239, 242, 251, 262, 271, 278, 284, 302, 311, 316, 319, 331, 341, 349, 358, 359, 361, 362, 379, 382, 398

Offset: 1

Henry Bottomley, Feb 23 2000

nonn

Conjecture: m is on this list iff m is an odd number all of whose factors are on this list or m is 2 or 4 times such an odd number.
A001176(a(n)) = A128924(a(n),1) = 1. - Reinhard Zumkeller, Jan 16 2014
Also numbers n such that A001175(n) = A001177(n). - Daniel Suteu, Aug 08 2018

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Brennan Benfield and Michelle Manes, The Fibonacci Sequence is Normal Base 10, arXiv:2202.08986 [math.NT], 2022.
Brennan Benfield and Oliver Lippard, Connecting Zeros in Pisano Periods to Prime Factors of K-Fibonacci Numbers, arXiv:2407.20048 [math.NT], 2024.
M. Renault, Fibonacci sequence modulo m

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.
                             |   m=1    |   m=2   |   m=3
-----------------------------+----------+---------+---------
The sequence {x(n)}          | A000045  | A000129 | A006190
The sequence {w(k)}          | A001176  | A214027 | A322906
Primes p such that w(p) = 1  | A112860* | A309580 | A309586
Primes p such that w(p) = 2  | A053027  | A309581 | A309587
Primes p such that w(p) = 4  | A053028  | A261580 | A309588
Numbers k such that w(k) = 1 | this seq | A309583 | A309591
Numbers k such that w(k) = 2 | A053030  | A309584 | A309592
Numbers k such that w(k) = 4 | A053029  | A309585 | A309593
* and also A053032 U {2}

a053031 n = a053031_list !! (n-1)
a053031_list = filter ((== 1) . a001176) [1..]
-- Reinhard Zumkeller, Jan 16 2014

With[{s = {1}~Join~Table[Count[Drop[NestWhile[Append[#, Mod[Total@ Take[#, -2], n]] &, {1, 1}, If[Length@ # < 3, True, Take[#, -2] != {1, 1}] &], -2], 0], {n, 2, 400}]}, Position[s, 1][[All, 1]] ] (* Michael De Vlieger, Aug 08 2018 *)

entryp(p)=my(k=p+[0, -1, 1, 1, -1][p%5+1], f=factor(k)); for(i=1, #f[, 1],for(j=1, f[i, 2], if((Mod([1, 1; 1, 0], p)^(k/f[i, 1]))[1, 2], break); k/=f[i, 1])); k
entry(n)=if(n==1, return(1)); my(f=factor(n), v); v=vector(#f~, i, if(f[i, 1]>1e14, entryp(f[i, 1]^f[i, 2]), entryp(f[i, 1])*f[i, 1]^(f[i, 2]-1))); if(f[1, 1]==2&&f[1, 2]>1, v[1]=3<Charles R Greathouse IV, Dec 14 2016

A112860 2 together with A053032.

Original entry on oeis.org

2, 11, 19, 29, 31, 59, 71, 79, 101, 131, 139, 151, 179, 181, 191, 199, 211, 229, 239, 251, 271, 311, 331, 349, 359, 379, 419, 431, 439, 461, 479, 491, 499, 509, 521, 541, 571, 599, 619, 631, 659, 691, 709, 719, 739, 751, 809, 811, 839, 859, 911, 919, 941, 971

Offset: 1

N. J. A. Sloane, Nov 30 2007

nonn

Consists of the primes that are in neither A053027 nor A053028.
From Jianing Song, Jun 16 2024: (Start)
Primes p such that A001176(p) = 1.
For p > 2, p is in this sequence if and only if A001175(p) == 2 (mod 4), and if and only if A001177(p) == 2 (mod 4). For a proof of the equivalence between A001176(p) = 1 and A001177(p) == 2 (mod 4), see Section 2 of my link below.
This sequence contains all primes congruent to 11, 19 (mod 20). This corresponds to case (3) for k = 3 in the Conclusion of Section 1 of my link below.
Conjecturely, this sequence has density 1/3 in the primes. (End) [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]

T. D. Noe, Table of n, a(n) for n=1..1001
C. Ballot and M. Elia, Rank and period of primes in the Fibonacci sequence; a trichotomy, Fib. Quart., 45 (No. 1, 2007), 56-63 (The sequence B1).
Jianing Song, Lucas sequences and entry point modulo p

Cf. A001175, A001177.
Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.
                             |   m=1     |   m=2   |   m=3
-----------------------------+-----------+---------+---------
The sequence {x(n)}          | A000045   | A000129 | A006190
The sequence {w(k)}          | A001176   | A214027 | A322906
Primes p such that w(p) = 1  | this seq* | A309580 | A309586
Primes p such that w(p) = 2  | A053027   | A309581 | A309587
Primes p such that w(p) = 4  | A053028   | A261580 | A309588
Numbers k such that w(k) = 1 | A053031   | A309583 | A309591
Numbers k such that w(k) = 2 | A053030   | A309584 | A309592
Numbers k such that w(k) = 4 | A053029   | A309585 | A309593
* and also A053032 U {2}

A261580 Primes p such that A214028(p) is odd.

Original entry on oeis.org

5, 13, 29, 37, 53, 61, 101, 109, 137, 149, 157, 173, 181, 197, 229, 269, 277, 293, 317, 349, 373, 389, 397, 421, 461, 509, 521, 541, 557, 569, 593, 613, 653, 661, 677, 701, 709, 733, 757, 773, 797, 821, 829, 853, 857, 877, 941, 953, 997, 1013, 1021, 1061, 1069

Offset: 1

Michel Marcus, Aug 25 2015

nonn

From Jianing Song, Aug 13 2019: (Start)
Primes p with 4 zeros in a fundamental period of A000129 mod p, that is, primes p such that A214027(p) = 4. For a proof of the equivalence between A214027(p) = 4 and A214028(p) being odd, see Section 2 of my link below.
For p > 2, p is in this sequence if and only if A175181(p) == 4 (mod 8).
This sequence contains all primes congruent to 5 modulo 8. This corresponds to case (1) for k = 6 in the Conclusion of Section 1 of my link below.
Conjecturely, since (k+2)/2 = 4 is a square, this sequence has density 7/24 in the primes; see the end of Section 1 of my link. (End) [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 20 2024]
The conjecture above is an analog of Hasse's result that the set {p prime : ord(2,p) is odd} has density 7/24 in the primes, where ord(a,m) is the multiplicative order of a modulo m; see A014663. - Jianing Song, Jun 26 2025

			The smallest Pell number divisible by the prime 5 has index 3, which is odd, so 5 is in the sequence.

Jianing Song, Table of n, a(n) for n = 1..1280
Bernadette Faye and Florian Luca, Pell Numbers whose Euler Function is a Pell Number, arXiv:1508.05714 [math.NT], 2015.
Jianing Song, Lucas sequences and entry point modulo p

Cf. A214028, A261581.
Cf. also A175181.
Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.
                             |   m=1    |   m=2    |   m=3
-----------------------------+----------+----------+---------
The sequence {x(n)}          | A000045  | A000129  | A006190
The sequence {w(k)}          | A001176  | A214027  | A322906
Primes p such that w(p) = 1  | A112860* | A309580  | A309586
Primes p such that w(p) = 2  | A053027  | A309581  | A309587
Primes p such that w(p) = 4  | A053028  | this seq | A309588
Numbers k such that w(k) = 1 | A053031  | A309583  | A309591
Numbers k such that w(k) = 2 | A053030  | A309584  | A309592
Numbers k such that w(k) = 4 | A053029  | A309585  | A309593
* and also A053032 U {2}

f[n_] := Block[{k = 1}, While[Mod[Simplify[((1 + Sqrt@ 2)^k - (1 - Sqrt@ 2)^k)/(2 Sqrt@ 2)], n] != 0, k++]; k]; Select[Prime@ Range@ 180, OddQ@ f@ # &] (* Michael De Vlieger, Aug 25 2015 *)

pell(n) = polcoeff(Vec(x/(1-2*x-x^2) + O(x^(n+1))), n);
z(n) = {k=1; while (pell(k) % n, k++); k;}
lista(nn) = {forprime(p=2, nn, if (z(p) % 2, print1(p, ", ")););}

forprime(p=2, 1100, if(A214027(p)==4, print1(p, ", "))) \\ Jianing Song, Aug 13 2019

A309580 Primes p with 1 zero in a fundamental period of A000129 mod p.

Original entry on oeis.org

2, 7, 23, 31, 41, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263, 271, 311, 313, 353, 359, 367, 383, 409, 431, 439, 457, 463, 479, 487, 503, 599, 607, 631, 647, 719, 727, 743, 751, 761, 809, 823, 839, 863, 887, 911, 919, 967, 983, 991, 1031, 1039, 1063, 1087, 1103, 1129, 1151, 1201, 1223, 1231, 1279

Offset: 1

Jianing Song, Aug 10 2019

nonn

Primes p such that A214027(p) = 1.
For p > 2, p is in this sequence if and only if A175181(p) == 2 (mod 4), and if and only if A214028(p) == 2 (mod 4). For a proof of the equivalence between A214027(p) = 1 and A214028(p) == 2 (mod 4), see Section 2 of my link below.
This sequence contains all primes congruent to 7 modulo 8. This corresponds to case (3) for k = 6 in the Conclusion of Section 1 of my link below.
Conjecturely, since (k+2)/2 = 4 is a square, this sequence has density 7/24 in the primes; see the end of Section 1 of my link. [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]
The conjecture above is an analog of Hasse's result that the set {p prime : ord(2,p) is odd} has density 7/24 in the primes, where ord(a,m) is the multiplicative order of a modulo m; see A014663. - Jianing Song, Jun 26 2025

Jianing Song, Table of n, a(n) for n = 1..1280
Jianing Song, Lucas sequences and entry point modulo p

Cf. A175181, A214028.
Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.
                             |   m=1    |   m=2    |   m=3
-----------------------------+----------+----------+---------
The sequence {x(n)}          | A000045  | A000129  | A006190
The sequence {w(k)}          | A001176  | A214027  | A322906
Primes p such that w(p) = 1  | A112860* | this seq | A309586
Primes p such that w(p) = 2  | A053027  | A309581  | A309587
Primes p such that w(p) = 4  | A053028  | A261580  | A309588
Numbers k such that w(k) = 1 | A053031  | A309583  | A309591
Numbers k such that w(k) = 2 | A053030  | A309584  | A309592
Numbers k such that w(k) = 4 | A053029  | A309585  | A309593
* and also A053032 U {2}

forprime(p=2, 1300, if(A214027(p)==1, print1(p, ", ")))

A309581 Primes p with 2 zeros in a fundamental period of A000129 mod p.

Original entry on oeis.org

3, 11, 17, 19, 43, 59, 67, 73, 83, 89, 97, 107, 113, 131, 139, 163, 179, 193, 211, 227, 233, 241, 251, 257, 281, 283, 307, 331, 337, 347, 379, 401, 419, 433, 443, 449, 467, 491, 499, 523, 547, 563, 571, 577, 587, 601, 617, 619, 641, 643, 659, 673, 683, 691

Offset: 1

Jianing Song, Aug 10 2019

nonn

Primes p such that A214027(p) = 2.
For p > 2, p is in this sequence if and only if 8 divides A175181(p), and if and only if 4 divides A214028(p). For a proof of the equivalence between A214027(p) = 2 and 4 dividing A214028(p), see Section 2 of my link below.
This sequence contains all primes congruent to 3 modulo 8. This corresponds to case (2) for k = 6 in the Conclusion of Section 1 of my link below.
Conjecturely, since (k+2)/2 = 4 is a square, this sequence has density 5/12 in the primes; see the end of Section 1 of my link. [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]
The conjecture above is an analog of Hasse's result that the set {p prime : ord(2,p) is odd} has density 7/24 in the primes, where ord(a,m) is the multiplicative order of a modulo m; see A014663. - Jianing Song, Jun 26 2025

Jianing Song, Table of n, a(n) for n = 1..1600
Jianing Song, Lucas sequences and entry point modulo p

Cf. A175181, A214028.
Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.
                             |   m=1    |   m=2    |   m=3
-----------------------------+----------+----------+---------
The sequence {x(n)}          | A000045  | A000129  | A006190
The sequence {w(k)}          | A001176  | A214027  | A322906
Primes p such that w(p) = 1  | A112860* | A309580  | A309586
Primes p such that w(p) = 2  | A053027  | this seq | A309587
Primes p such that w(p) = 4  | A053028  | A261580  | A309588
Numbers k such that w(k) = 1 | A053031  | A309583  | A309591
Numbers k such that w(k) = 2 | A053030  | A309584  | A309592
Numbers k such that w(k) = 4 | A053029  | A309585  | A309593
* and also A053032 U {2}

forprime(p=2, 700, if(A214027(p)==2, print1(p, ", ")))

cp's OEIS Frontend

A001176 Number of zeros in fundamental period of Fibonacci numbers mod n.

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A053028 Odd primes p with 4 zeros in any period of the Fibonacci numbers mod p.

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A053032 Odd primes p with one zero in Fibonacci numbers mod p.

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A053027 Odd primes p with 2 zeros in Fibonacci numbers mod p.

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A053029 Numbers with 4 zeros in Fibonacci numbers mod m.

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A053031 Numbers with 1 zero in Fibonacci numbers mod m.

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A112860 2 together with A053032.

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A261580 Primes p such that A214028(p) is odd.

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