A081264 -id:A081264 - cp's OEIS frontend

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

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

A069107 Composite numbers k that divide Fibonacci(k+1).

Original entry on oeis.org

323, 377, 2834, 3827, 5777, 6479, 10877, 11663, 18407, 19043, 20999, 23407, 25877, 27323, 34943, 35207, 39203, 44099, 47519, 50183, 51983, 53663, 60377, 65471, 75077, 78089, 79547, 80189, 81719, 82983, 84279, 84419, 86063, 90287, 94667

Offset: 1

Benoit Cloitre, Apr 06 2002

Primes p congruent to +2 or -2 (mod 5) divide Fibonacci(p+1) (cf. A003631 and [Hardy and Wright]).

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Fifth edition), Oxford Univ. Press (Clarendon), 1979, Chap. X, p. 150.

Chai Wah Wu, Table of n, a(n) for n = 1..2000 (n = 1..250 from Reinhard Zumkeller, n = 251..1000 from Giovanni Resta)

Cf. A045468, A003631, A064739, A081264 (Fibonacci pseudoprimes).

Cf. A000045, A010051, A023172, A069104.

a069107 n = a069107_list !! (n-1)
a069107_list = h 2 $ drop 3 a000045_list where
   h n (fib:fibs) = if fib `mod` n > 0 || a010051 n == 1
       then h (n+1) fibs else n : h (n+1) fibs
-- Reinhard Zumkeller, Oct 13 2011

Select[Range[2,100000],!PrimeQ[#]&&Divisible[Fibonacci[#+1],#]&] (* Harvey P. Dale, Sep 18 2011 *)

is(n)=((Mod([1,1;1,0],n))^(n+1))[1,2]==0 && !isprime(n) && n>1 \\ Charles R Greathouse IV, Oct 07 2016

Fibonacci(2*a(n)) mod a(n) = a(n) - 1. - Gary Detlefs, May 26 2014

Corrected by Ralf Stephan, Oct 17 2002

A069106 Composite numbers k such that k divides F(k-1) where F(j) are the Fibonacci numbers.

Original entry on oeis.org

442, 1891, 2737, 4181, 6601, 6721, 8149, 13201, 13981, 15251, 17119, 17711, 30889, 34561, 40501, 51841, 52701, 64079, 64681, 67861, 68101, 68251, 78409, 88601, 88831, 90061, 96049, 97921, 115231, 118441, 138601, 145351, 146611, 150121, 153781, 163081, 179697, 186961, 191351, 194833

Offset: 1

Benoit Cloitre, Apr 06 2002

Primes p congruent to 1 or 4 (mod 5) divide F(p-1) (cf. A045468 and [Hardy and Wright]).

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Fifth edition), Oxford Univ. Press (Clarendon), 1979, Chap. X, p. 150.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Subsequence of A123976.
Cf. A045468, A003631, A064739, A081264 (Fibonacci pseudoprimes).
Cf. A002808, A000045.

#include  #include  #define STARTN 10 #define N_OF_MILLER_RABIN_TESTS 5 int main() { mpz_t n, f1, f2; int flag=0; /* flag? 0: f1 contains current F[n-1] 1: f2 = F[n-1] */ mpz_set_ui (n, STARTN); mpz_init (f1); mpz_init (f2); mpz_fib2_ui (f1, f2, STARTN-1); for (;;) { if (mpz_probab_prime_p (n, N_OF_MILLER_RABIN_TESTS)) goto next_iter; if (mpz_divisible_p (!flag? f1:f2, n)) { mpz_out_str (stdout, 10, n); printf (" "); fflush (stdout); } next_iter: mpz_add_ui (n, n, 1); mpz_add (!flag? f2:f1, f1, f2); flag = !flag; } }

a069106 n = a069106_list !! (n-1)
a069106_list = [x | x <- a002808_list, a000045 (x-1) `mod` x == 0]
-- Reinhard Zumkeller, Jul 19 2013

A069106[nn_] := Select[Complement[Range[2,nn],Prime[Range[2,PrimePi[ nn]]]],Divisible[ Fibonacci[ #-1],#]&] (* Harvey P. Dale, Jul 05 2011 *)

fibmod(n,m)=((Mod([1,1;1,0],m))^n)[1,2]
is(n)=!isprime(n) && !fibmod(n-1,n) && n>1 \\ Charles R Greathouse IV, Oct 06 2016

Corrected and extended (with C program) by Ralf Stephan, Oct 13 2002

a(35)-a(40) added by Reinhard Zumkeller, Jul 19 2013

A141137 Even Fibonacci pseudoprimes: even composite numbers k such that either (1) k divides Fibonacci(k-1) if k mod 5 = 1 or -1 or (2) k divides Fibonacci(k+1) if k mod 5 = 2 or -2.

Original entry on oeis.org

8539786, 12813274, 17340938, 33940178, 64132426, 89733106, 95173786, 187473826, 203211098, 234735586, 353686906, 799171066, 919831058, 1188287794, 1955272906, 2166139898, 2309861746, 2864860298, 3871638242, 5313594466, 5867301826

Offset: 1

T. D. Noe, Jun 09 2008

These even Fibonacci pseudoprimes (FPPs) were found by Kenny Richardson (kenyai(AT)yahoo.com). See A081264 for odd FPPs and references. Be aware that some authors use the term "Fibonacci pseudoprime" for pseudoprimes in Lucas sequences. For example, see A005845 for Lucas V(1,-1) pseudoprimes.

a(69) > 2.6 * 10^11. - Dana Jacobsen, May 25 2015

Dana Jacobsen, Table of n, a(n) for n = 1..68
Dorin Andrica and Ovidiu Bagdasar, Recurrent Sequences: Key Results, Applications, and Problems, Springer (2020), p. 88.
Dorin Andrica and Ovidiu Bagdasar, On Generalized Lucas Pseudoprimality of Level k, Mathematics (2021) Vol. 9, 838.

Cf. A081264.

use ntheory ":all"; for (3..1e10) { my $n = $<<1; $e = (0,-1,1,1,-1)[$n%5]; next unless $e; say $n unless (lucas_sequence($n, 1, -1, $n+$e))[0]; } # _Dana Jacobsen, May 25 2015

a(19) from Giovanni Resta, Jul 20 2013

a(20)-a(21) from Dana Jacobsen, May 25 2015

A212424 Frobenius pseudoprimes with respect to Fibonacci polynomial x^2 - x - 1.

Original entry on oeis.org

4181, 5777, 6721, 10877, 13201, 15251, 34561, 51841, 64079, 64681, 67861, 68251, 75077, 90061, 96049, 97921, 100127, 113573, 118441, 146611, 161027, 162133, 163081, 186961, 197209, 219781, 231703, 252601, 254321, 257761, 268801, 272611

Offset: 1

Max Alekseyev, May 16 2012

nonn

Grantham incorrectly claims that "the first Frobenius pseudoprime with respect to the Fibonacci polynomial x^2 - x - 1 is 5777". Crandall and Pomerance state that the first such Frobenius pseudoprime is actually 4181.
The Frobenius (1,-1) pseudoprimes are a subset of the odd Fibonacci pseudoprimes A081264. Among other ways, this can be seen by Theorem 3.6.6 of Crandall and Pomerance (2005) where the Frobenius criterion with respect to x^2 - Px + Q is an additional condition on an input which has passed the Lucas test for the same polynomial. - Dana Jacobsen, Aug 05 2015
Many other quadratics have a sparser set of pseudoprimes.  For example, while there are 98702 pseudoprimes below 10^13 with respect to the Fibonacci polynomial, there are only 3897 for x^2 - 3x - 5. - Dana Jacobsen, Aug 05 2015
This is the intersection of A049062 and (A081264 union A141137), that is, composite k coprime to 5 such that Fibonacci(k) == (k/5) (mod k) and that k divides Fibonacci(k-(k/5)), where (k/5) is the Legendre or Jacobi symbol. - Jianing Song, Sep 12 2018

R. Crandall, C. B. Pomerance. Prime Numbers: A Computational Perspective. Springer, 2nd ed., 2005.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (from Dana Jacobsen's site, terms 1..653 from Max Alekseyev)
Dorin Andrica and Ovidiu Bagdasar, Recurrent Sequences: Key Results, Applications, and Problems, Springer (2020), p. 89.
Jon Grantham, Frobenius pseudoprimes, Mathematics of Computation 70 (234): 873-891, 2001. doi: 10.1090/S0025-5718-00-01197-2.
Dana Jacobsen, Pseudoprime Statistics, Tables, and Data (includes terms through 10^13)
A. Rotkiewicz, Lucas and Frobenius Pseudoprimes, Annales Mathematicae Silesiane, 17 (2003): 17-39.
Lawrence Somer, Lucas sequences {Uk} for which U2p and Up are pseudoprimes for almost all primes p, Fibonacci Quart. 44 (2006), no. 1, 7-12.
Eric W. Weisstein, Frobenius Pseudoprime, MathWorld.

Cf. A049062, A081264, A141137.
Cf. also A005845, A094063, A094395, A094411.
Terms congruent to 2 or 3 mod 5 are given in A212423.

{ isFP(n) = if(ispseudoprime(n),return(0)); t=Mod(x*Mod(1,n),(x^2-x-1)*Mod(1,n))^n; (kronecker(5,n)==-1 && t==1-x)||(kronecker(5,n)==1 && t==x) }

use ntheory ":all"; foroddcomposites { say if is_frobenius_pseudoprime($,1,-1) } 1e10; # _Dana Jacobsen, Aug 05 2015

A340095 Odd composite integers m such that A052918(m-J(m,29)) == 0 (mod m) and gcd(m,29)=1, where J(m,29) is the Jacobi symbol.

Original entry on oeis.org

9, 15, 27, 45, 91, 121, 135, 143, 1547, 1573, 1935, 2015, 6543, 6721, 8099, 10403, 10877, 10905, 13319, 13741, 13747, 14399, 14705, 16109, 16471, 18901, 19043, 19109, 19601, 19951, 20591, 22753, 24639, 26599, 26937, 27593

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 the identity
U(p-J(p,D)) == 0 (mod p) when p is prime, b=-1 and D=a^2+4.
This sequence contains the odd composite integers with U(m-J(m,D)) == 0 (mod m).
For a=5 and b=-1, we have D=29 and U(m) recovers A052918(m).
If even numbers greater than 2 that are coprime to 29 are allowed, then 26, 442, 6994, ... would also be terms. - Jianing Song, Jan 09 2021

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

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.
D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math., 18, 47 (2021).
D. Andrica and O. Bagdasar, On generalized pseudoprimality of level k, Mathematics 2021, 9(8), 838.

Cf. A052918, A071904, A081264 (a=1, b=-1), A327653 (a=3, b=-1), A340096 (a=7, b=-1), A340097 (a=3, b=1), A340098 (a=5, b=1), A340099 (a=7, b=1).

Select[Range[3,28000, 2], CoprimeQ[#, 29] && CompositeQ[#] && Divisible[Fibonacci[#-JacobiSymbol[#, 29], 5], #] &]

Coprime condition added to definition by Georg Fischer, Jul 20 2022

A340096 Odd composite integers m such that A054413(m-J(m,53)) == 0 (mod m), where J(m,53) is the Jacobi symbol.

Original entry on oeis.org

25, 35, 51, 65, 91, 175, 325, 391, 455, 575, 1247, 1295, 1633, 1763, 1775, 1921, 2275, 2407, 2599, 2651, 3367, 4199, 4579, 4623, 5629, 6441, 9959, 10465, 10825, 10877, 12025, 13021, 15155, 16021, 18881, 19019, 19039, 19307, 19669

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 the identity
U(p-J(p,D)) == 0 (mod p) when p is prime, b=-1 and D=a^2+4.
This sequence contains the odd composite integers with U(m-J(m,D)) == 0 (mod m).
For a=7 and b=-1, we have D=53 and U(m) recovers A054413(m).
If even numbers greater than 2 that are coprime to 53 are allowed, then 10, 50, 370, 5050, ... would also be terms. - Jianing Song, Jan 09 2021

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. A054413, A071904, A081264 (a=1, b=-1), A327653 (a=3,b=-1), A340095 (a=5, b=-1)

Cf. A340097 (a=3, b=1), A340098 (a=5, b=1), A340099 (a=7, b=1).

Select[Range[3,20000, 2], CoprimeQ[#, 53] && CompositeQ[#] && Divisible[Fibonacci[#-JacobiSymbol[#, 53], 7], #] &]

A340097 Odd composite integers m such that A001906(m-J(m,5)) == 0 (mod m) and gcd(m,5)=1, where J(m,5) is the Jacobi symbol.

Original entry on oeis.org

21, 323, 329, 377, 451, 861, 1081, 1819, 1891, 2033, 2211, 3653, 3827, 4089, 4181, 5671, 5777, 6601, 6721, 8149, 8557, 10877, 11309, 11663, 13201, 13861, 13981, 14701, 15251, 17119, 17513, 17711, 17941, 18407, 19043, 19951, 20473, 23407, 25369, 25651, 25877, 27323, 27511

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 the identity
U(p-J(p,D)) == 0 (mod p) when p is prime, b=1 and D=a^2-4.
This sequence contains the odd composite integers with U(m-J(m,D)) == 0 (mod m).
For a=3 and b=1, we have D=5 and U(m) recovers A001906(m).

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

D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math., 18, 47 (2021).
D. Andrica and O. Bagdasar, On generalized pseudoprimality of level k, Mathematics 2021, 9(8), 838.
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. A001906, A071904, A081264 (a=1, b=-1), A327653 (a=3,b=-1), A340095 (a=5, b=-1), A340096 (a=7, b=-1), A340098 (a=5, b=1), A340099 (a=7, b=1).

Select[Range[3, 30000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[ChebyshevU[# - JacobiSymbol[#, 5] - 1, 3/2], #] &]

Coprime condition added to definition by Georg Fischer, Jul 20 2022

A340098 Odd composite integers m such that A004254(m-J(m,21)) == 0 (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.

Original entry on oeis.org

115, 253, 391, 527, 551, 713, 715, 779, 935, 1705, 1807, 1919, 2627, 2893, 2929, 3281, 4033, 4141, 5191, 5671, 5777, 5983, 6049, 6479, 7645, 7739, 8695, 9361, 11663, 11815, 12121, 12209, 12265, 14491, 17249, 17963, 18299, 18407, 20087, 20099, 21505, 22499, 24463

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 the identity
U(p-J(p,D)) == 0 (mod p) when p is prime, b=1 and D=a^2-4.
This sequence contains the odd composite integers with U(m-J(m,D)) == 0 (mod m).
For a=5 and b=1, we have D=21 and U(m) recovers A004254(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. A004254, A071904, A081264 (a=1, b=-1), A327653 (a=3,b=-1), A340095 (a=5, b=-1), A340096 (a=7, b=-1), A340097 (a=3, b=1), A340099 (a=7, b=1).

Select[Range[3, 25000, 2], CoprimeQ[#, 21] && CompositeQ[#] && Divisible[ChebyshevU[# - JacobiSymbol[#, 21] - 1, 5/2], #] &]

A340099 Odd composite integers m such that A004187(m-J(m,45)) == 0 (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol.

Original entry on oeis.org

323, 329, 377, 451, 1081, 1771, 1819, 1891, 2033, 3653, 3827, 4181, 5671, 5777, 6601, 6721, 7471, 7931, 8149, 8557, 10877, 11309, 11663, 13201, 13861, 13981, 14701, 15251, 15449, 17119, 17513, 17687, 17711, 17941, 18407, 19043, 19951, 20447, 20473, 23407, 23771, 23851, 23999

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 the identity
U(p-J(p,D)) == 0 (mod p) when p is prime, b=1 and D=a^2-4.
This sequence contains the odd composite integers with U(m-J(m,D)) == 0 (mod m).
For a=7 and b=1, we have D=45 and U(m) recovers A004187(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. A004187, A071904, A081264 (a=1, b=-1), A327653 (a=3,b=-1), A340095 (a=5, b=-1), A340096 (a=7, b=-1), A340097 (a=3, b=1), A340098 (a=5, b=1).

Select[Range[3, 25000, 2], CoprimeQ[#, 45] && CompositeQ[#] && Divisible[ChebyshevU[# - JacobiSymbol[#, 45] - 1, 7/2], #] &]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A069107 Composite numbers k that divide Fibonacci(k+1).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A069106 Composite numbers k such that k divides F(k-1) where F(j) are the Fibonacci numbers.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

C

Haskell

Mathematica

PARI

Extensions

A141137 Even Fibonacci pseudoprimes: even composite numbers k such that either (1) k divides Fibonacci(k-1) if k mod 5 = 1 or -1 or (2) k divides Fibonacci(k+1) if k mod 5 = 2 or -2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Perl

Extensions

A212424 Frobenius pseudoprimes with respect to Fibonacci polynomial x^2 - x - 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

PARI

Perl

A340095 Odd composite integers m such that A052918(m-J(m,29)) == 0 (mod m) and gcd(m,29)=1, where J(m,29) is the Jacobi symbol.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Extensions

A340096 Odd composite integers m such that A054413(m-J(m,53)) == 0 (mod m), where J(m,53) is the Jacobi symbol.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs