A005845 -id:A005845 - cp's OEIS frontend

A335671 Odd composite integers m such that A087130(m) == 5 (mod m).

9, 27, 65, 121, 145, 377, 385, 533, 1035, 1189, 1305, 1885, 2233, 2465, 4081, 5089, 5993, 6409, 6721, 7107, 10877, 11281, 11285, 13281, 13369, 13741, 13833, 14705, 15457, 16721, 17545, 18901, 19601, 19951, 20329, 20705, 22881, 24769, 25345, 26599, 26937, 28741, 29161

Offset: 1

Ovidiu Bagdasar, Jun 17 2020

nonn

If p is a prime, then A087130(p)==5 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
The generalized Pell-Lucas sequence of integer parameters (a,b) defined by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a, satisfy the identity V(p)==a (mod p) whenever p is prime and b=-1,1.
For a=5, b=-1, V(n) recovers A087130(n).

			9 is the first odd composite integer for which A087130(9)=2744420==5 (mod 9).

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

Robert Israel, Table of n, a(n) for n = 1..1000
D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, preprint for Mediterr. J. Math. 18, 47 (2021).

Cf. A006497, A005845 (a=1), A330276 (a=2), A335669 (a=3), A335670 (a=4).

M:= <<5|1>,<1|0>>:
f:= proc(n) uses LinearAlgebra:-Modular;
local A;
A:= Mod(n,M,integer[8]);
A:= MatrixPower(n,A,n);
2*A[1,1] - 5*A[1,2] mod n;
end proc:
select(t -> f(t) = 5 and not isprime(t), [seq(i,i=3..10^5,2)]); # Robert Israel, Jun 19 2020

Select[Range[3, 30000, 2], CompositeQ[#] && Divisible[LucasL[#, 5] - 5, #] &] (* Amiram Eldar, Jun 18 2020 *)

More terms from Jinyuan Wang, Jun 17 2020

A217255 Strong Lucas pseudoprimes.

Original entry on oeis.org

5459, 5777, 10877, 16109, 18971, 22499, 24569, 25199, 40309, 58519, 75077, 97439, 100127, 113573, 115639, 130139, 155819, 158399, 161027, 162133, 176399, 176471, 189419, 192509, 197801, 224369, 230691, 231703, 243629, 253259, 268349, 288919, 313499, 324899

Offset: 1

Robert Baillie, Mar 16 2013

nonn

Strong Lucas pseudoprimes with parameters (P, Q) defined by Selfridge's Method A.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (from Dana Jacobsen's site, terms 1..2000 from R. J. Mathar)
Martin R. Albrecht, Jake Massimo, Kenneth G. Paterson, Juraj Somorovsky, Prime and Prejudice: Primality Testing Under Adversarial Conditions, Proceedings of the 2018 ACM SIGSAC Conference on Computer and Communications Security, 281-298.
Robert Baillie and Samuel S. Wagstaff, Jr., Lucas Pseudoprimes, Mathematics of Computation, 35 (1980), 1391-1417.
Robert Baillie, Mathematica program for A217255
Dana Jacobsen, Pseudoprime Statistics, Tables, and Data (includes terms through 10^14)

Cf. A217120 (Lucas pseudoprimes as defined by Baillie and Wagstaff).
Cf. A005845 (Lucas pseudoprimes as defined by Bruckman).
Cf. A217719 (extra strong Lucas pseudoprimes as defined by Baillie).

Mathematica
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A337625 Odd composite integers m such that F(m)^2 == 1 (mod m) and L(m) == 1 (mod m), where F(m) and L(m) are the m-th Fibonacci and Lucas numbers, respectively.

Original entry on oeis.org

2737, 4181, 5777, 6721, 10877, 13201, 15251, 29281, 34561, 51841, 64079, 64681, 67861, 68251, 75077, 80189, 90061, 96049, 97921, 100127, 105281, 113573, 118441, 146611, 161027, 162133, 163081, 179697, 186961, 194833, 197209, 219781, 228241, 231703, 252601, 254321

Offset: 1

Ovidiu Bagdasar, Sep 19 2020

nonn

Intersection of A005845 and A337231.
These numbers may be called weak generalized Fibonacci-Lucas-Bruckner pseudoprimes.
If p is a prime, then F(p)^2 == 1 (mod p) and L(p) == 1 (mod p).
This sequence contains the odd composite integers for which these congruences hold.
For a,b integers, the following sequences are defined:
generalized Lucas sequences by U(n+2)=a*U(n+1)-b*U(n) and U(0)=0, U(1)=1,
generalized Pell-Lucas sequences by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a.
These satisfy the identities U(p)^2 == 1 and V(p)==a (mod p) for p prime and b=1,-1.
These numbers may be called weak generalized Lucas-Bruckner pseudoprimes of parameters a and b.The current sequence is defined for a=1 and b=-1.
Examples: a(n) is also the number of Jones graphs on n nodes.

Dorin Andrica and Ovidiu Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math., 18 (2021), 47.

Cf. A005845 and A337231.

Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 1]*Fibonacci[#, 1] - 1, #] && Divisible[LucasL[#, 1] - 1, #] &]

More terms from Amiram Eldar, Sep 19 2020

A094398 Numbers k that divide Lucas(k) + 1.

Original entry on oeis.org

1, 2, 4, 8, 15, 16, 32, 64, 128, 256, 323, 368, 512, 736, 1024, 1472, 2048, 2944, 4096, 4879, 5655, 5888, 6479, 7055, 8192, 8464, 9879, 10815, 11663, 11776, 12935, 16384, 16928, 18407, 19043, 23407, 23552, 31535, 32768, 33856, 34943, 35207, 35296

Offset: 1

Eric Rowland, May 01 2004

nonn

The powers of 2 (A000079) are in the sequence. - Michel Lagneau, Feb 09 2015

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

Cf. A000079, A000204, A005845, A094402.

Select[Range[40000],Divisible[LucasL[#]+1,#]&]  (* Harvey P. Dale, Apr 13 2011 *)

is(n)=(Mod([0,1;1,1],n)^n*[2;1])[1,1]==-1 \\ Charles R Greathouse IV, Nov 04 2016

A213060 Lucas(n) mod n, Lucas(n)= A000032(n).

Original entry on oeis.org

0, 1, 1, 3, 1, 0, 1, 7, 4, 3, 1, 10, 1, 3, 14, 15, 1, 0, 1, 7, 11, 3, 1, 2, 11, 3, 22, 7, 1, 18, 1, 31, 4, 3, 4, 34, 1, 3, 17, 7, 1, 18, 1, 7, 41, 3, 1, 2, 29, 23, 4, 7, 1, 0, 44, 47, 4, 3, 1, 22, 1, 3, 41, 63, 11, 18, 1, 7, 50, 53, 1, 2, 1, 3, 64, 7, 73, 18

Offset: 1

Gary Detlefs, Jun 03 2012

a(n) = 1 for all prime values of n. Composite values for which a(n) = 1 are listed in A005845.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000032, A005845.

Cf. A002708 (Fibonacci(n) mod n).

[Lucas(n) mod (n) : n in [1..120]]; // Vincenzo Librandi, Nov 19 2015

with(combinat):f:=n-> fibonacci(n):L:=n->f(2*n)/f(n): seq(L(n) mod n, n= 1..75)
# alternative
A213060 := proc(n::integer)
    modp(A000032(n),n) ;
end proc:
seq(A213060(n),n=1..100) ; # R. J. Mathar, Oct 02 2019

Table[Mod[LucasL[n], n], {n, 100}] (* T. D. Noe, Jun 06 2012 *)

A225876 Composite n which divide s(n)+1, where s is the linear recurrence sequence s(n) = -s(n-1) + s(n-2) - s(n-3) + s(n-5) with initial terms (5, -1, 3, -7, 11).

Original entry on oeis.org

4, 14791044, 143014853, 253149265, 490434564, 600606332, 993861182, 3279563483

Offset: 1

Matt McIrvin, May 23 2013

The pseudoprimes derived from the fifth-order linear recurrence A225984(n) are analogous to the Perrin pseudoprimes A013998, and the Lucas pseudoprimes A005845.
For prime p, A225984(p) == p - 1 (mod p). The pseudoprimes are composite numbers satisfying the same relation. 4 = 2^2; 14791044 = 2^2 * 3 * 19 * 29 * 2237; 143014853 = 907 * 157679.
Like the Perrin test, the modular sequence is periodic so simple pre-tests can be performed.  Numbers divisible by 2, 3, 4, 5, 9, and 25 have periods 31, 11, 62, 24, 33, and 120 respectively. - Dana Jacobsen, Aug 29 2016
a(9) > 1.4*10^11. - Dana Jacobsen, Aug 29 2016

			A225984(4) = 11, and 11 == 3 (mod 4). Since 4 is composite, it is a pseudoprime with respect to A225984.

K. Brown, Proof of Generalized Little Theorem of Fermat, proves that for prime p, a(p) == a(1) (mod p) for recurrences of the form of A225984.
R. Holmes, comments to M. McIrvin's post on Google+ (found terms 4 through 7)

N=10^10;
default(primelimit, N);
M = [0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1; 1, 0, -1, 1, -1];
a(n)=lift( trace( Mod(M, n)^n ) );
ta(n)=lift( trace( Mod(M, n) ) );
{ for (n=2, N,
    if ( isprime(n), next() );
    if ( a(n)==ta(n), print1(n, ", "); );
); }
/* Matt McIrvin, after Joerg Arndt's program for A013998, May 23 2013 */

Terms 4 through 7 found by Richard Holmes, added by Matt McIrvin, May 27 2013

a(8) from Dana Jacobsen, Aug 29 2016

A164824 Lucas pseudoprimes whose reversal is prime.

Original entry on oeis.org

10877, 97921, 113573, 163081, 186961, 302101, 1106327, 1149851, 1317121, 1392169, 1533601, 1653601, 1690501, 3218801, 3338221, 3399527, 3967201, 7405201, 7640137, 7678321, 9264097, 9439201, 9532033, 9582145, 10237921

Offset: 1

Jonathan Vos Post, Aug 27 2009

			a(1) = 10877 = 73 * 149 because that is the 8th Lucas pseudoprime, and its reversal 77801 is prime.

Cf. A000040, A004086, A005845, A006567, A007500.

A005845 INTERSECTION (A006567 UNION A007500). {n in A005845 such that A004086(n) is in A000040}. {n such that n | (L_n - 1), where n is composite and L_n = Lucas numbers A000032, and R(n) is prime}.

More terms from R. J. Mathar, Sep 27 2009

A178375 The greatest common prime divisor of A000032(n)-1 and A001608(n), or 1 if no such divisor exists.

Original entry on oeis.org

2, 3, 2, 5, 1, 7, 2, 3, 1, 11, 1, 13, 1, 1, 2, 17, 1, 19, 1, 1, 2, 23, 1, 5, 1, 3, 1, 29, 1, 31, 2, 7, 1, 3, 1, 37, 1, 7, 1, 41, 1, 43, 2, 1, 2, 47, 1, 7, 2, 1, 3, 53, 1, 7, 1, 1, 2, 59, 1, 61, 1, 1, 2, 7, 1, 67, 3, 1, 1, 71, 1, 73, 2, 1, 1, 5, 1, 79, 1, 7, 1, 83, 1, 2, 2, 7, 2, 89, 1

Offset: 2

Vladimir Shevelev, May 26 2010

nonn

If n is prime, then n divides c(n). If n is composite and divides c(n) it is a pseudoprime to both the Lucas (Bruckman) and Perrin tests, which is the intersection of A005845 and A013998.

Conjecture: Records of the sequence are consecutive primes.

Cf. A000032, A001608, A178380, A005845, A013998.

More terms from R. J. Mathar, Aug 08 2010

A338078 Odd composite integers m such that A085447(m) == 6 (mod m).

Original entry on oeis.org

57, 185, 385, 481, 629, 721, 779, 1121, 1441, 1729, 2419, 2737, 5665, 6721, 7471, 8401, 9361, 10465, 10561, 11285, 11521, 11859, 12257, 13585, 14705, 15281, 16321, 16583, 18849, 24721, 25345, 25441, 25593, 30745, 33649, 35219, 36481, 36581, 37949, 38665, 39169

Offset: 1

Ovidiu Bagdasar, Oct 08 2020

nonn

If p is a prime, then A085447(p)==6 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
The generalized Pell-Lucas sequence of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy the identity V(p)==a (mod p) whenever p is prime and b=-1,1.
For a=6, b=-1, V(m) recovers A085447(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)

Cf. A006497, A005845 (a=1), A330276 (a=2), A335669 (a=3), A335670 (a=4), A335671 (a=5).

Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[LucasL[#, 6] - 6, #] &]

More terms from Amiram Eldar, Oct 09 2020

A338079 Odd composite integers m such that A086902(m) == 7 (mod m).

Original entry on oeis.org

25, 51, 91, 161, 265, 325, 425, 561, 791, 1105, 1113, 1325, 1633, 1921, 1961, 2001, 2465, 2599, 2651, 2737, 3445, 4081, 4505, 4929, 7345, 7685, 8449, 9361, 10325, 10465, 10825, 11285, 11713, 12025, 12291, 13021, 15457, 17111, 18193, 18881, 18921, 19307

Offset: 1

Ovidiu Bagdasar, Oct 08 2020

nonn

If p is a prime, then A086902(p)==7 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
The generalized Pell-Lucas sequence of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy the identity V(p)==a (mod p) whenever p is prime and b=-1,1.
For a=7, b=-1, V(m) recovers A086902(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)

Cf. A006497, A005845 (a=1), A330276 (a=2), A335669 (a=3), A335670 (a=4), A335671 (a=5), A338078 (a=6).

Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[LucasL[#, 7] - 7, #] &]

cp's OEIS Frontend

A335671 Odd composite integers m such that A087130(m) == 5 (mod m).

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A217255 Strong Lucas pseudoprimes.

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A337625 Odd composite integers m such that F(m)^2 == 1 (mod m) and L(m) == 1 (mod m), where F(m) and L(m) are the m-th Fibonacci and Lucas numbers, respectively.

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Mathematica

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A094398 Numbers k that divide Lucas(k) + 1.

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PARI

A213060 Lucas(n) mod n, Lucas(n)= A000032(n).

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Magma

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Mathematica

A225876 Composite n which divide s(n)+1, where s is the linear recurrence sequence s(n) = -s(n-1) + s(n-2) - s(n-3) + s(n-5) with initial terms (5, -1, 3, -7, 11).

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PARI

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A164824 Lucas pseudoprimes whose reversal is prime.

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A178375 The greatest common prime divisor of A000032(n)-1 and A001608(n), or 1 if no such divisor exists.

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A338078 Odd composite integers m such that A085447(m) == 6 (mod m).