A335669 -id:A335669 - cp's OEIS frontend

A335670 Odd composite integers m such that A014448(m) == 4 (mod m).

9, 85, 161, 341, 705, 897, 901, 1105, 1281, 1853, 2465, 2737, 3745, 4181, 4209, 4577, 5473, 5611, 5777, 6119, 6721, 9701, 9729, 10877, 11041, 12209, 12349, 13201, 13481, 14981, 15251, 16185, 16545, 16771, 19669, 20591, 20769, 20801, 21845, 23323, 24465, 25345

Offset: 1

Ovidiu Bagdasar, Jun 17 2020

nonn

If p is a prime, then A014448(p)==4 (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=4, b=-1, V(n) recovers A014448(n) (even Lucas numbers).

			9 is the first odd composite integer for which A014448(9)=439204==4 (mod 9).

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

Chai Wah Wu, 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), A335671 (a=5).

M:= <<4|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] - 4*A[1,2] mod n;
end proc:
select(t -> f(t) = 4 and not isprime(t), [seq(i,i=3..10^5,2)]); # Robert Israel, Jun 19 2020

Select[Range[3, 25000, 2], CompositeQ[#] && Divisible[LucasL[3#] - 4, #] &] (* Amiram Eldar, Jun 18 2020 *)

More terms from Jinyuan Wang, Jun 17 2020

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

Original entry on oeis.org

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

A337626 Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 3 (mod m), where U(m) and V(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=3 and b=-1, respectively.

Original entry on oeis.org

33, 119, 385, 561, 649, 1189, 1441, 2065, 2289, 2465, 2849, 4187, 6545, 12871, 13281, 14041, 16109, 18241, 22049, 23479, 24769, 25345, 28421, 31631, 34997, 38121, 38503, 41441, 45961, 48577, 50545, 53585, 56279, 58081, 59081, 61447, 63393, 66385, 75077, 91187

Offset: 1

Ovidiu Bagdasar, Sep 19 2020

nonn

Intersection of A335669 and A337234.
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=3 and b=-1.

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

Cf. A335669, A337234.

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

More terms from Amiram Eldar, Sep 19 2020

A335672 Odd composite integers m such that A005248(m) == 3 (mod m).

Original entry on oeis.org

15, 105, 195, 231, 323, 377, 435, 665, 705, 1443, 1551, 1891, 2465, 2737, 2849, 3289, 3689, 3745, 3827, 4181, 4465, 4879, 5655, 5777, 6479, 6601, 6721, 7055, 7743, 8149, 9879, 10815, 10877, 11305, 11395, 11663, 12935, 13201, 13981, 15251, 15301, 17119, 17261, 17711, 18407, 18915, 19043, 20999

Offset: 1

Ovidiu Bagdasar, Jun 17 2020

nonn

If p is a prime, then A005248(p)==3 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
The generalized Pell-Lucas sequences 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=3, b=1, V(n) recovers A005248(n) (bisection of Fibonacci numbers).

			15 is the first odd composite integer for which A005248(15)=18604984==3 (mod 15).

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

Chai Wah Wu, 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. A005248, A335669 (a=3,b=-1), A335673 (a=4,b=1), A335674 (a=5,b=1).

Select[Range[3, 10000, 2], CompositeQ[#] && Divisible[LucasL[2#] - 3, #] &] (* Amiram Eldar, Jun 18 2020 *)

A335673 Composite integers m such that A003500(m) == 4 (mod m).

Original entry on oeis.org

10, 209, 230, 231, 399, 430, 455, 530, 901, 903, 923, 989, 1295, 1729, 1855, 2015, 2211, 2345, 2639, 2701, 2795, 2911, 3007, 3201, 3439, 3535, 3801, 4823, 5291, 5719, 6061, 6767, 6989, 7421, 8569, 9503, 9591, 9869, 9890, 10439, 10609, 11041, 11395, 11951, 11991

Offset: 1

Ovidiu Bagdasar, Jun 17 2020

nonn

If p is a prime, then A003500(p)==4 (mod p).
This sequence contains the composite integers for which the congruence holds.
The generalized Pell-Lucas sequences 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=4, b=1, V(n)=A003500(n).

			m=10 is the first composite integer for which A003500(m)==4 (mod m).

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

Amiram Eldar, Table of n, a(n) for n = 1..10000 (first 1000 odd terms from Chai Wah Wu)
D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, preprint for Mediterr. J. Math. 18, 47 (2021).

Cf. A005248, A335669 (a=3,b=-1), A335672 (a=3,b=1), A335674 (a=5,b=1).

A330206 is the subsequence of odd terms.

Select[Range[3, 20000], CompositeQ[#] && Divisible[Round@LucasL[2#, Sqrt[2]] - 4, #] &] (* Amiram Eldar, Jun 18 2020 *)

my(M=[1,2;1,3]); forcomposite(m=5, 10^5, if(trace(Mod(M,m)^m)==4, print1(m,", "))); \\ Joerg Arndt, Jun 18 2020

More terms from Joerg Arndt, Jun 18 2020

A335674 Odd composite integers m such that A003501(m) == 5 (mod m).

Original entry on oeis.org

15, 21, 35, 105, 161, 195, 255, 345, 385, 399, 465, 527, 551, 609, 741, 897, 1105, 1295, 1311, 1807, 1919, 2001, 2015, 2071, 2085, 2121, 2415, 2737, 2915, 3289, 3815, 4031, 4033, 4355, 4879, 4991, 5291, 5777, 5983, 6049, 6061, 6083, 6479, 6601, 6785, 7645, 7905, 8695, 8855, 8911, 9361, 9591, 9889

Offset: 1

Ovidiu Bagdasar, Jun 17 2020

nonn

If p is a prime, then A003501(p)==5 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
The generalized Pell-Lucas sequences 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 A003501(n).

			15 is the first odd composite integer for which the relation A003501(15)=16098445550==5 (mod 15) holds.

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

Chai Wah Wu, 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. A005248, A335669 (a=3,b=-1), A335672 (a=3,b=1), A335673 (a=4,b=1).

Select[Range[3, 5000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 5/2] - 5, #] &] (* Amiram Eldar, Jun 18 2020 *)

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, #] &]

A338082 Odd composite integers m such that A056854(m) == 7 (mod m).

Original entry on oeis.org

9, 15, 21, 35, 45, 63, 99, 105, 195, 231, 315, 323, 329, 369, 377, 423, 435, 451, 595, 665, 705, 805, 861, 903, 1081, 1189, 1443, 1551, 1819, 1833, 1869, 1891, 1935, 2033, 2211, 2345, 2465, 2737, 2849, 2871, 2961, 3059, 3289, 3653, 3689, 3745, 3827, 4005, 4059

Offset: 1

Ovidiu Bagdasar, Oct 08 2020

nonn

If p is a prime, then A056854(p)==7 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
The generalized Pell-Lucas sequences 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) when p is prime and b=-1,1.
For a=7 and b=1, V(m) recovers A056854(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. A005248, A335669 (a=3,b=-1), A335672 (a=3,b=1), A335673 (a=4,b=1), A335674 (a=5,b=1), A337233 (a=6,b=1).

Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 7/2] - 7, #] &]
Select[Range[9,5001,2],CompositeQ[#]&&Mod[LucasL[4#],#]==7&] (* Harvey P. Dale, Apr 28 2022 *)

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A335670 Odd composite integers m such that A014448(m) == 4 (mod m).

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

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A337626 Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 3 (mod m), where U(m) and V(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=3 and b=-1, respectively.

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A335672 Odd composite integers m such that A005248(m) == 3 (mod m).

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A335673 Composite integers m such that A003500(m) == 4 (mod m).

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A335674 Odd composite integers m such that A003501(m) == 5 (mod m).

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

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