A006190 -id:A006190 - cp's OEIS frontend

A322906 The number of zeros in the fundamental Pisano period of the 3-Fibonacci numbers A006190 modulo n.

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

Offset: 1

Jianing Song, Jan 05 2019

nonn

a(n) is the multiplicative order of A006190(A322907(n)+1) modulo n.
a(n) has value 1, 2 or 4. This is because A006190(k,m+1)^4 == 1 (mod A006190(k,m)).
Conjecture: For primes p == 1, 9, 17, 25, 49, 81 (mod 104), the probability of a(p^e) taking on the value 1, 2, 4 is 1/6, 2/3, 1/6, respectively; for primes p == 29, 53, 61, 69, 77, 101 (mod 104), the probability of a(p^e) taking on the value 1, 4 is 1/2, 1/2, respectively.

Jianing Song, Table of n, a(n) for n = 1..1000

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = k*x(n+1) + x(n). Then the periods, ranks and the ratios of the periods to the ranks modulo a given integer n are given by:
k = 1: A001175 (periods), A001177 (ranks), A001176 (ratios).
k = 2: A175181 (periods), A214028 (ranks), A214027 (ratios).
k = 3: A175182 (periods), A322907 (ranks), this sequence (ratios).

A006190(m) = ([3, 1; 1, 0]^m)[2, 1]
a(n) = my(i=1); while(A006190(i)%n!=0, i++); znorder(Mod(A006190(i+1), n))

For n > 2, T(n,k) = 4 iff A322907(n) is odd; 1 iff A322907(n) is even but not divisible by 4; 2 iff A322907(n) is divisible by 4.
For primes p == 3, 23, 27, 35, 43, 51 (mod 52), a(p^e) = 1.
For primes p == 5, 21, 33, 37, 41, 45 (mod 52), a(p^e) = 4.
For primes p == 7, 11, 15, 19, 31, 47 (mod 52), a(p^e) = 2.
a(13^e) = 4. a(2^e) = 1 if e = 1, 2 and 2 if e >= 3.
a(n) = A175182(n)/A322907(n).

A309586 Primes p with 1 zero in a fundamental period of A006190 mod p.

Original entry on oeis.org

2, 3, 23, 43, 53, 61, 79, 101, 103, 107, 127, 131, 139, 173, 179, 191, 199, 211, 251, 263, 277, 283, 311, 347, 367, 419, 433, 439, 443, 467, 491, 503, 523, 547, 563, 569, 571, 599, 607, 647, 659, 677, 719, 727, 751, 757, 823, 829, 859, 881, 883, 887, 907

Offset: 1

Jianing Song, Aug 10 2019

nonn

Primes p such that A322906(p) = 1.
For p > 2, p is in this sequence if and only if A175182(p) == 2 (mod 4), and if and only if A322907(p) == 2 (mod 4). For a proof of the equivalence between A322906(p) = 1 and A322907(p) == 2 (mod 4), see Section 2 of my link below.
This sequence contains all primes congruent to 3, 23, 27, 35, 43, 51 modulo 52. This corresponds to case (3) for k = 11 in the Conclusion of Section 1 of my link below.
Conjecturely, this sequence has density 1/3 in the primes. [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]

Jianing Song, Table of n, a(n) for n = 1..1200
Jianing Song, Lucas sequences and entry point modulo p

Cf. A175182, A322907.
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 | this seq
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, 900, if(A322906(p)==1, print1(p, ", ")))

A309587 Primes p with 2 zeros in a fundamental period of A006190 mod p.

Original entry on oeis.org

7, 11, 17, 19, 31, 47, 59, 67, 71, 83, 113, 151, 163, 167, 223, 227, 239, 257, 271, 307, 313, 331, 337, 359, 379, 383, 431, 463, 479, 487, 499, 521, 587, 601, 619, 631, 641, 643, 673, 683, 691, 739, 743, 787, 809, 811, 827, 839, 863, 947, 967, 983

Offset: 1

Jianing Song, Aug 10 2019

nonn

Primes p such that A322906(p) = 2.
For p > 2, p is in this sequence if and only if 8 divides A175182(p), and if and only if 4 divides A322907(p). For a proof of the equivalence between A322906(p) = 2 and 4 dividing A322907(p), see Section 2 of my link below.
This sequence contains all primes congruent to 7, 11, 15, 19, 31, 47 modulo 52. This corresponds to case (2) for k = 11 in the Conclusion of Section 1 of my link below.
Conjecturely, this sequence has density 1/3 in the primes. [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]

Jianing Song, Table of n, a(n) for n = 1..1200
Jianing Song, Lucas sequences and entry point modulo p

Cf. A006190, A175182, A322907.
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 | this seq
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, 1000, if(A322906(p)==2, print1(p, ", ")))

A309588 Primes p with 4 zeros in a fundamental period of A006190 mod p.

Original entry on oeis.org

5, 13, 29, 37, 41, 73, 89, 97, 109, 137, 149, 157, 181, 193, 197, 229, 233, 241, 269, 281, 293, 317, 349, 353, 373, 389, 397, 401, 409, 421, 449, 457, 461, 509, 541, 557, 577, 593, 613, 617, 653, 661, 701, 709, 733, 761, 769, 773, 797, 821, 853, 857, 877

Offset: 1

Jianing Song, Aug 10 2019

nonn

Primes p such that A322906(p) = 4.
For p > 2, p is in this sequence if and only if A175182(p) == 4 (mod 8), and if and only if A322907(p) is odd. For a proof of the equivalence between A322906(p) = 4 and A322907(p) being odd, see Section 2 of my link below.
This sequence contains all primes congruent to 5, 21, 33, 37, 41, 45 modulo 52. This corresponds to case (1) for k = 11 in the Conclusion of Section 1 of my link below.
Conjecturely, this sequence has density 1/3 in the primes. [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]

Jianing Song, Table of n, a(n) for n = 1..1200
Jianing Song, Lucas sequences and entry point modulo p

Cf. A006190, A175182, A322907.
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 | this seq
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, 900, if(A322906(p)==4, print1(p, ", ")))

A309591 Numbers k with 1 zero in a fundamental period of A006190 mod k.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 12, 18, 23, 27, 36, 43, 46, 53, 54, 61, 69, 79, 81, 86, 92, 101, 103, 106, 107, 108, 122, 127, 129, 131, 138, 139, 158, 159, 162, 172, 173, 179, 183, 191, 199, 202, 206, 207, 211, 212, 214, 237, 243, 244, 251, 254, 258, 262, 263, 276

Offset: 1

Jianing Song, Aug 10 2019

nonn

Numbers k such that A322906(k) = 1.

The odd numbers k satisfy A175182(k) == 2 (mod 4).

Jianing Song, Table of n, a(n) for n = 1..3000

Cf. A175182.
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 | this seq
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}

for(k=1, 300, if(A322906(k)==1, print1(k, ", ")))

A309592 Numbers k with 2 zeros in a fundamental period of A006190 mod k.

Original entry on oeis.org

7, 8, 11, 14, 15, 16, 17, 19, 20, 21, 22, 24, 28, 30, 31, 32, 33, 34, 35, 38, 39, 40, 42, 44, 45, 47, 48, 49, 51, 52, 55, 56, 57, 59, 60, 62, 63, 64, 66, 67, 68, 70, 71, 72, 75, 76, 77, 78, 80, 83, 84, 85, 87, 88, 90, 91, 93, 94, 95, 96, 98, 99, 100, 102, 104

Offset: 1

Jianing Song, Aug 10 2019

nonn

Numbers k such that A322906(k) = 2.
This sequence contains all numbers k such that 4 divides A322907(k). As a consequence, this sequence contains all numbers congruent to 7, 11, 15, 19, 31, 47 modulo 52.
This sequence contains all odd numbers k such that 8 divides A175182(k).

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A175182, A322907.
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 | this seq
Numbers k such that w(k) = 4 | A053029  | A309585 | A309593
* and also A053032 U {2}

for(k=1, 100, if(A322906(k)==2, print1(k, ", ")))

A309593 Numbers k with 4 zeros in a fundamental period of A006190 mod k.

Original entry on oeis.org

5, 10, 13, 25, 26, 29, 37, 41, 50, 58, 65, 73, 74, 82, 89, 97, 109, 125, 130, 137, 145, 146, 149, 157, 169, 178, 181, 185, 193, 194, 197, 205, 218, 229, 233, 241, 250, 269, 274, 281, 290, 293, 298, 314, 317, 325, 338, 349, 353, 362, 365, 370, 373, 377

Offset: 1

Jianing Song, Aug 10 2019

nonn

Numbers k such that A322906(k) = 4.

Also numbers k such that A214027(k) is odd.

Jianing Song, Table of n, a(n) for n = 1..2000

Cf. A322907.
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 | this seq
* and also A053032 U {2}

for(k=1, 400, if(A322906(k)==4, print1(k, ", ")))

A175182 Pisano period of the 3-Fibonacci numbers A006190.

Original entry on oeis.org

1, 3, 2, 6, 12, 6, 16, 12, 6, 12, 8, 6, 52, 48, 12, 24, 16, 6, 40, 12, 16, 24, 22, 12, 60, 156, 18, 48, 28, 12, 64, 48, 8, 48, 48, 6, 76, 120, 52, 12, 28, 48, 42, 24, 12, 66, 96, 24, 112, 60, 16, 156, 26, 18, 24, 48, 40, 84, 24, 12, 30, 192, 48, 96, 156, 24, 136, 48, 22, 48, 144

Offset: 1

R. J. Mathar, Mar 01 2010

Period of the sequence defined by reading A006190 modulo n.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Sergio Falcon and Ángel Plaza, k-Fibonacci sequences modulo m, Chaos, Solit. Fractals 41 (2009), 497-504.
Eric Weisstein's World of Mathematics, Pisano period.
Wikipedia, Pisano period.

Cf. A001175, A006190, A175181, A175183, A175184, A175185.

F := proc(k,n) option remember; if n <= 1 then n; else k*procname(k,n-1)+procname(k,n-2) ; end if; end proc:
Pper := proc(k,m) local cha, zer,n,fmodm ; cha := [] ; zer := [] ; for n from 0 do fmodm := F(k,n) mod m ; cha := [op(cha),fmodm] ; if fmodm = 0 then zer := [op(zer),n] ; end if; if nops(zer) = 5 then break; end if; end do ; if [op(1..zer[2],cha) ] = [ op(zer[2]+1..zer[3],cha) ] and [op(1..zer[2],cha)] = [ op(zer[3]+1..zer[4],cha) ] and [op(1..zer[2],cha)] = [ op(zer[4]+1..zer[5],cha) ] then return zer[2] ; elif [op(1..zer[3],cha) ] = [ op(zer[3]+1..zer[5],cha) ] then return zer[3] ; else return zer[5] ; end if; end proc:
k := 3 ; seq( Pper(k,m),m=1..80) ;

Table[s = t = Mod[{0, 1}, n]; cnt = 1; While[tmp = Mod[3*t[[2]] + t[[1]], n]; t[[1]] = t[[2]]; t[[2]] = tmp; s!= t, cnt++]; cnt, {n, 100}] (* Vincenzo Librandi, Dec 20 2012, T. D. Noe *)

A327653 Composite numbers k coprime to 13 such that k divides A006190(k-Kronecker(13,k)).

Original entry on oeis.org

10, 119, 649, 1189, 1763, 3599, 4187, 5559, 6681, 12095, 12403, 12685, 12871, 12970, 14041, 14279, 15051, 16109, 19043, 22847, 23479, 24769, 26795, 28421, 30743, 30889, 31631, 31647, 33919, 34997, 37949, 38503, 39203, 41441, 46079, 48577, 49141, 50523, 50545, 53301, 56279, 58081, 58589

Offset: 1

Jianing Song, Sep 20 2019

nonn

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n) = m*x(n-1) + x(n-2) for k >= 2. For primes p, we have (a) p divides x(p-((m^2+4)/p)); (b) x(p) == ((m^2+4)/p) (mod p), where (D/p) is the Kronecker symbol. This sequence gives composite numbers k such that gcd(k, m^2+4) = 1 and that a condition similar to (a) holds for k, where m = 3.
If k is not required to be coprime to m^2 + 4 (= 13), then there are 360 such k <= 10^5 and 1506 such k <= 10^6, while there are only 62 terms <= 10^5 and 197 terms <= 10^6 in this sequence.
Also composite numbers k coprime to 13 such that A322907(k) divides k - Kronecker(13,k).

			A006190(9) = 12970 which is divisible by 10, so 10 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

             m                      m=1            m=2      m=3
k | x(k-Kronecker(m^2+4,k))*  A081264 U A141137  A327651  this seq
k | x(k)-Kronecker(m^2+4,k)        A049062       A099011  A327654
            both                   A212424       A327652  A327655
* k is composite and coprime to m^2 + 4.
Cf. A006190, A322907, A011583 ({Kronecker(13,n)}).

seqmod(n, m)=((Mod([3, 1; 1, 0], m))^n)[1, 2]
isA327653(n)=!isprime(n) && !seqmod(n-Kronecker(13,n), n) && gcd(n,13)==1 && n>1

A322907 Entry points for the 3-Fibonacci numbers A006190.

Original entry on oeis.org

1, 3, 2, 6, 3, 6, 8, 6, 6, 3, 4, 6, 13, 24, 6, 12, 8, 6, 20, 6, 8, 12, 22, 6, 15, 39, 18, 24, 7, 6, 32, 24, 4, 24, 24, 6, 19, 60, 26, 6, 7, 24, 42, 12, 6, 66, 48, 12, 56, 15, 8, 78, 26, 18, 12, 24, 20, 21, 12, 6, 30, 96, 24, 48, 39, 12, 68, 24, 22, 24, 72, 6

Offset: 1

Jianing Song, Jan 05 2019

nonn

a(n) is the smallest k > 0 such that n divides A006190(k).
a(n) is also called the rank of A006190(n) modulo n.
For primes p == 1, 9, 17, 25, 29, 49 (mod 52), a(p) divides (p - 1)/2.
For primes p == 3, 23, 27, 35, 43, 51 (mod 52), a(p) divides p - 1, but a(p) does not divide (p - 1)/2.
For primes p == 5, 21, 33, 37, 41, 45 (mod 52), a(p) divides (p + 1)/2.
For primes p == 7, 11, 15, 19, 31, 47 (mod 52), a(p) divides p + 1, but a(p) does not divide (p + 1)/2.
a(n) <= (12/7)*n for all n, where the equality holds if and only if n = 2*7^e, e >= 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Jianing Song)

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = k*x(n+1) + x(n). Then the periods, ranks and the ratios of the periods to the ranks modulo a given integer n are given by:
k = 1: A001175 (periods), A001177 (ranks), A001176 (ratios).
k = 2: A175181 (periods), A214028 (ranks), A214027 (ratios).
k = 3: A175182 (periods), this sequence (ranks), A322906 (ratios).
Cf. A006190.

A006190(m) = ([3, 1; 1, 0]^m)[2, 1]
a(n) = my(i=1); while(A006190(i)%n!=0, i++); i

a(m*n) = a(m)*a(n) if gcd(m, n) = 1.
For odd primes p, a(p^e) = p^(e-1)*a(p) if p^2 does not divide a(p). Any counterexample would be a 3-Wall-Sun-Sun prime.
a(2^e) = 3 if e = 1, 6 if e = 2 and 3*2^(e-2) if e >= 3. a(13^e) = 13^e, e >= 1.

cp's OEIS Frontend

A322906 The number of zeros in the fundamental Pisano period of the 3-Fibonacci numbers A006190 modulo n.

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A309586 Primes p with 1 zero in a fundamental period of A006190 mod p.

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A309587 Primes p with 2 zeros in a fundamental period of A006190 mod p.

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A309588 Primes p with 4 zeros in a fundamental period of A006190 mod p.

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A309591 Numbers k with 1 zero in a fundamental period of A006190 mod k.

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A309592 Numbers k with 2 zeros in a fundamental period of A006190 mod k.

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A309593 Numbers k with 4 zeros in a fundamental period of A006190 mod k.

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A175182 Pisano period of the 3-Fibonacci numbers A006190.

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A327653 Composite numbers k coprime to 13 such that k divides A006190(k-Kronecker(13,k)).

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