A006190 -id:A006190 - cp's OEIS frontend

A337234 Odd composite integers m such that A006190(m)^2 == 1 (mod m).

9, 33, 55, 63, 99, 119, 153, 231, 385, 399, 561, 649, 935, 981, 1023, 1071, 1179, 1189, 1199, 1441, 1595, 1763, 1881, 1953, 2001, 2065, 2255, 2289, 2465, 2703, 2751, 2849, 2871, 3519, 3599, 3655, 3927, 4059, 4081, 4187, 5015, 5151, 5559, 6061, 6119, 6215, 6273, 6431

Offset: 1

Ovidiu Bagdasar, Aug 20 2020

nonn

If p is a prime, then A006190(p)^2 == 1 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
The generalized Lucas sequence of integer parameters (a,b) defined by U(n+2) = a*U(n+1)-b*U(n) and U(0)=0, U(1)=1, satisfies the identity U^2(p) == 1 (mod p) whenever p is prime and b=-1.
For a=3, b=-1, U(n) recovers A006190(n) ("Bronze" Fibonacci numbers).

D. Andrica and O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).

Dorin Andrica and Ovidiu Bagdasar, On Generalized Lucas Pseudoprimality of Level k, Mathematics (2021) Vol. 9, 838.
D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, preprint for Mediterr. J. Math. 18, 47 (2021).

Cf. A337231 (a=1, odd terms), A337232 (a=1, even terms), A337233 (a=2).

Select[Range[3, 25000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 3]*Fibonacci[#, 3] - 1, #] &]

A327654 Composite numbers k coprime to 13 such that k divides A006190(k) - Kronecker(13,k).

Original entry on oeis.org

4, 8, 9, 119, 399, 649, 1023, 1179, 1189, 1199, 1881, 2703, 3519, 4081, 4187, 5151, 7055, 7361, 10349, 12871, 13833, 14041, 15519, 16109, 18639, 22593, 23479, 24769, 26937, 28421, 29007, 31631, 34111, 34997, 38503, 41441, 44671, 48577, 50545, 51711, 53823, 56279, 57407, 58081, 59081

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 (b) holds for k, where m = 3.

If k is not required to be coprime to m^2 + 4 (= 13), then there are 352 such k <= 10^5, and 1457 such k <= 10^6, while there are only 54 terms <= 10^5 and 148 terms <= 10^6 in this sequence.

			A006190(8) = 3927 == Kronecker(13,8) (mod 8), so 8 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..2000

             m                      m=1            m=2      m=3
k | x(k-Kronecker(m^2+4,k))*  A081264 U A141137  A327651  A327653
k | x(k)-Kronecker(m^2+4,k)        A049062       A099011  this seq
            both                   A212424       A327652  A327655
* k is composite and coprime to m^2 + 4.
Cf. A006190, A011583 ({Kronecker(13,n)}).

seqmod(n, m)=((Mod([3, 1; 1, 0], m))^n)[1, 2]
isA327654(n)=!isprime(n) && seqmod(n, n)==kronecker(13,n) && gcd(n,13)==1 && n>1

A337235 Even composite integers m such that A006190(m)^2 == 1 (mod m).

Original entry on oeis.org

4, 8, 16, 68, 1208, 1424, 3056, 3824, 3928, 20912, 52174, 63716, 88708, 123148, 161872, 582224, 887566, 17083292, 18900412, 34648888, 39991684, 44884912, 51390736, 103170448, 107825236, 132238514, 279900272, 686071244, 769252508, 3251623346, 3358311986, 3535011826

Offset: 1

Ovidiu Bagdasar, Aug 20 2020

nonn

If p is a prime, then A006190(p)^2 == 1 (mod p).
This sequence contains the even composite integers for which the congruence holds.
The generalized Lucas sequence of integer parameters (a,b) defined by U(n+2) = a*U(n+1)-b*U(n) and U(0)=0, U(1)=1, satisfies the identity U^2(p) == 1 (mod p) whenever p is prime and b=-1,1.
For a=3, b=-1, U(n) recovers A006190(n) ("Bronze" Fibonacci numbers).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).

D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, preprint for Mediterr. J. Math. 18, 47 (2021).

Cf. A337231 (a=1, odd terms), A337232 (a=1, even terms), A337233 (a=2), A337234 (a=3, odd terms).

Select[Range[3, 25000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 5]*Fibonacci[#, 5] - 1, #] &]

More terms from Amiram Eldar, Aug 21 2020

a(18)-a(32) from Daniel Suteu, Aug 29 2020

A340119 Odd composite integers m such that A006190(2*m-J(m,13)) == 1 (mod m), where J(m,13) is the Jacobi symbol.

Original entry on oeis.org

9, 27, 63, 81, 99, 119, 153, 243, 567, 649, 729, 759, 891, 903, 1071, 1189, 1377, 1431, 1539, 1763, 1881, 1953, 2133, 2187, 3599, 3897, 4187, 4585, 5103, 5313, 5559, 5589, 5819, 6561, 6681, 6831, 6993, 8019, 8127, 8829, 8855, 9639, 9999, 10611, 11135, 11691, 11961

Offset: 1

Ovidiu Bagdasar, Dec 28 2020

nonn

The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy U(2*p-J(p,D)) == 1 (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4. The composite integers m with the property U(k*m-J(m,D)) == U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k- and parameter a. Here b=-1, a=3, D=13 and k=2, while U(m) is A006190(m).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 2018, 24(1), 9--15.

Cf. A006190, A071904, A081264 (a=1, b=-1, k=1), A327653 (a=3, b=-1, k=1).

Cf. A340118 (a=1, b=-1, k=2), A340120 (a=5, b=-1, k=2), A340121 (a=7, b=-1, k=2).

Select[Range[3, 12000, 2], CoprimeQ[#, 13] && CompositeQ[#] && Divisible[Fibonacci[2*#-JacobiSymbol[#, 13], 3] - 1, #] &]

A209493 Indices of primes in sequence A006190.

Original entry on oeis.org

2, 5, 11, 17, 61, 103, 167, 193, 293, 643, 647, 911, 11243, 29437, 55021, 80141

Offset: 1

Vaclav Kotesovec, Mar 09 2012

Index must be prime. The indices greater than 293 yield probable primes.

Cf. A006190, A006497, A000129, A096650.

seq=RecurrenceTable[{a[n]==3*a[n-1]+a[n-2], a[1]==1, a[2]==3}, a, {n, 1000}]; Select[Range[1000], PrimeQ[seq[[#]]]&]

a(14)-a(15) from Vaclav Kotesovec, Sep 08 2013

a(16) from Michael S. Branicky, Nov 03 2024

A340236 Odd composite integers m such that A006190(3*m-J(m,13)) == 3 (mod m), where J(m,13) is the Jacobi symbol.

Original entry on oeis.org

9, 119, 121, 187, 327, 345, 649, 705, 1003, 1089, 1121, 1189, 1881, 2091, 2299, 3553, 4187, 5461, 5565, 5841, 6165, 6485, 7107, 7139, 7145, 7467, 7991, 8321, 8449, 11041, 12705, 12871, 13833, 14041, 16109, 16851

Offset: 1

Ovidiu Bagdasar, Jan 01 2021

nonn

The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy U(3*p-J(p,D)) == a (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4. The composite integers m with the property U(k*m-J(m,D)) == U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k- and parameter a.

Here b=-1, a=3, D=13 and k=3, while U(m) is A006190(m).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 2018, 24(1), 9--15.

Cf. A006190, A071904, A327653 (a=3, b=-1, k=1), A340119 (a=3, b=-1, k=2).

Cf. A340235 (a=1, b=-1, k=3), A340237 (a=5, b=-1, k=3), A340238 (a=7, b=-1, k=3).

Select[Range[3, 15000, 2], CoprimeQ[#, 13] && CompositeQ[#] && Divisible[Fibonacci[3*#-JacobiSymbol[#, 13], 3] - 3, #] &]

A253247 Pisano period of A006190(n^2) divided by Pisano period of A006190(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 7, 8, 9, 5, 11, 4, 13, 7, 5, 16, 17, 9, 19, 10, 21, 11, 23, 8, 25, 13, 27, 7, 29, 5, 31, 32, 33, 17, 35, 36, 37, 19, 39, 40, 41, 7, 43, 11, 45, 23, 47, 16, 49, 25, 51, 26, 53, 27, 55, 14, 57, 29, 59, 10, 61, 31, 63, 64, 65, 11, 67, 17, 69, 35, 71, 72

Offset: 1

Eric Chen, Apr 09 2015

nonn

For all n, a(n)|n.
Conjecture: a(n) = 1 only for n = 1 and 6. (This conjecture is true if and only if the generalized Wall's conjecture to A006190 is true.)
If there exists any prime p such that A175182(p^2) = A175182(p), then the conjecture fails.
For any prime p, these three statements are equivalent:
(1) A175182(p^2) = A175182(p).
(2) A006190(p-(p|13)) = 3 (mod p^2).
(3) A006497(p) = 1 (mod p^2).
Since A175182(241^2) = A175182(241) = 484, so the prime 241 is a Wall-Sun-Sun prime to A006190 (Lucas (P, Q) = (3, -1)) and no others < 10^8, so the conjecture is true for all primes < 10^8 except 241.
All of Wall's theorems are true for A175182. For example, let P(n) = A175182(n), p and q are primes, then P(pq) = lcm(P(p), P(q)), and for every prime p, P(p)|(p-1) if (p|13) = 1, P(p)|(2p+2) if (p|13) = -1 (P(13) = 52, which if divisible by 13), while (p|13) is the Legendre symbol, and the fixed points of A175182 are 1, 6, and 12*13^k, k>0.

Eric Chen, Table of n, a(n) for n = 1..1000

Cf. A237517, A175182.

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^2) div Pper(k, m), m=1..300) ;

A006190[n_] := Fibonacci[n, 3];
A175182[n_] := Module[{k=1}, While[Mod[A006190[k], n] != 0 || Mod[A006190[k+1]-1, n] != 0, k++]; k];
Table[A175182[n^2] / A175182[n], {n, 72}] (* corrected by Jason Yuen, Jun 28 2025 *)

fibmod(n, m)=((Mod([3, 1; 1, 0], m))^n)[1, 2]
entry_p(p)=my(k=1, c=Mod(1, p), o); while(c, [o, c]=[c, 3*c+o]; k++); k
entry(n)=if(n==1, return(1)); my(f=factor(n), v); v=vector(#f~, i, if(f[i, 1]>1e8 && f[i, 1] != 241, entry_p(f[i, 1]^f[i, 2]), entry_p(f[i, 1])*f[i, 1]^(f[i, 2] - 1))); if(f[1, 1]==2&&f[1, 2]>1, v[1]=3<

a(n) = A175182(n^2) / A175182(n).

A253807 Primitive part of A006190(n), n >= 1.

Original entry on oeis.org

1, 3, 10, 11, 109, 12, 1189, 119, 1297, 131, 141481, 118, 1543321, 1429, 15445, 14159, 183642229, 1299, 2003229469, 14041, 1837837, 170039, 238367471761, 14158, 23854956949, 1854841, 2186871697, 1670761, 309400794703549

Offset: 1

Wolfdieter Lang, Jan 19 2015

A006190(n) = Product_{k divides n} a(k), n >= 1.

Cf. A000010, A006190, A008683, A013595.

Cf. A061446, A008555, A105603.

(* b = A006190 *) b[0] = 0; b[1] = 1; b[n_] := b[n] = 3*b[n-1] + b[n-2]; a[n_] := Product[b[d]^MoebiusMu[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jan 20 2015 *)

a(n) = ((3-sqrt(13))/2)^phi(n)*cyclotomic(n, -(11 - 3*sqrt13)/2) for n >= 1 and a(1) = 1, where phi is Euler's totient A000010 and the coefficient table for the cyclotomic polynomials is given in A013595.

a(n) = Product_{d|n} A006190(d)^mu(n/d), where mu = A008683, n >= 1.

A270997 Numbers k such that k | A006190(k-1).

Original entry on oeis.org

1, 3, 10, 17, 23, 29, 33, 43, 53, 61, 79, 101, 103, 107, 113, 127, 131, 139, 157, 173, 179, 181, 191, 199, 211, 233, 251, 257, 263, 269, 277, 283, 311, 313, 337, 347, 367, 373, 385, 389, 419, 433, 439, 443, 467, 491, 503, 521, 523, 547, 561, 563, 569, 571, 599, 601, 607, 641, 647, 649, 653, 659

Offset: 1

Altug Alkan, Mar 28 2016

nonn

This sequence appears to generate many prime numbers.
The first few composite terms in this sequence are 10, 33, 385, 561, 649, ...
Contains all members of A038883 except 13. - Robert Israel, Jun 03 2019
That is, contains all primes which are congruent to  +-1, +-3 or +-4 (mod 13). - M. F. Hasler, Feb 16 2022

			10 is a term because A006190(9) = 12970 is divisible by 10.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006190, A038883, A270951.

M:= <<3,1>|<1,0>>:
filter:= proc(n) uses LinearAlgebra[Modular];
  local A;
  A:= Mod(n,M,integer);
  MatrixPower(n,A,n-1)[1,2]=0
end proc:
filter(1):= true:
select(filter, [$1..659]); # Robert Israel, Jun 03 2019

nn = 660; s = LinearRecurrence[{3, 1}, {0, 1}, nn]; Select[Range@ nn, Divisible[s[[#]], #] &](* Michael De Vlieger, Mar 28 2016, after Harvey P. Dale at A006190 *)

a006190(n) = ([1, 3; 1, 2]^n)[2, 1];
for(n=1, 1e3, if(Mod(a006190(n-1), n) == 0, print1(n, ", ")));

A308948 a(n) = A006190(A322907(n)+1) mod n.

Original entry on oeis.org

0, 1, 1, 1, 3, 1, 6, 5, 1, 3, 10, 1, 8, 13, 4, 9, 16, 1, 18, 9, 13, 21, 1, 13, 18, 5, 1, 13, 12, 19, 30, 17, 10, 33, 6, 1, 31, 37, 25, 29, 32, 13, 1, 21, 19, 1, 46, 25, 48, 43, 16, 25, 1, 1, 21, 41, 37, 17, 58, 49, 1, 61, 55, 33, 18, 43, 66, 33, 1, 41, 70, 37

Offset: 1

Jianing Song, Jul 02 2019

nonn

A322907(n) is the smallest k > 0 such that n divides A006190(k).
Let M = [{3, 1}, {1, 0}], I = [{1, 0}, {0, 1}] is the 2 X 2 identity matrix, then A322907(n) is the smallest k > 0 such that M^k == r*I (mod n) for some r such that 0 <= r < n, and a(n) gives the value r.
A322906(n) is the multiplicative order of a(n) modulo n, which can only take value 1, 2 or 4.

			For n = 7, {A006190(n) mod 7 : n > 0} = {1, 3, 3, 5, 4, 3, 6, 0, 6, ...}, so a(7) = 6. Also, A322907(7) = 8, and M^8 mod 7 = [{6, 0}, {0, 6}], so a(7) = 6.

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

Cf. A006190, A322906, A322907.

Similar sequences: A217036, A308947.

a[n_] := For[k = 1, True, k++, If[Divisible[Fibonacci[k, 3], n], Return[ Mod[Fibonacci[k + 1, 3], n]]]];
Array[a, 100] (* Jean-François Alcover, Jul 05 2019 *)

a(n) = my(M=[3, 1; 1, 0]); for(k=1, 12*n/7, if((Mod(M,n)^k)[2,1]==0, return(lift((Mod(M,n)^k)[1,1]))))

Also a(n) = A006190(A322907(n)-1) mod n.
a(2^e) = 1 if e = 1, 2, 2^(e-1) + 1 if e >= 3; a(p^e) = a(p)^(p^(e-1)) mod p^e for odd primes p.
For odd primes p, a(p^e) = 1 if and only if A322907(p) == 2 (mod 4); a(p^e) = p^e - 1 if and only if 4 divides A322907(p).

cp's OEIS Frontend

A337234 Odd composite integers m such that A006190(m)^2 == 1 (mod m).

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A327654 Composite numbers k coprime to 13 such that k divides A006190(k) - Kronecker(13,k).

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A337235 Even composite integers m such that A006190(m)^2 == 1 (mod m).

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A340119 Odd composite integers m such that A006190(2*m-J(m,13)) == 1 (mod m), where J(m,13) is the Jacobi symbol.

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A209493 Indices of primes in sequence A006190.

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A340236 Odd composite integers m such that A006190(3*m-J(m,13)) == 3 (mod m), where J(m,13) is the Jacobi symbol.

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A253247 Pisano period of A006190(n^2) divided by Pisano period of A006190(n).

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A253807 Primitive part of A006190(n), n >= 1.

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