A340097 -id:A340097 - cp's OEIS frontend

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.

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], #] &]

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], #] &]

A340122 Odd composite integers m such that A001906(2*m-J(m,5)) == J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

Original entry on oeis.org

9, 21, 27, 63, 81, 189, 243, 323, 329, 351, 377, 423, 451, 567, 729, 783, 861, 891, 963, 1081, 1701, 1743, 1819, 1891, 1967, 2033, 2187, 2211, 2871, 2889, 2961, 3321, 3653, 3807, 3827, 4089, 4181, 5103, 5229, 5671, 5777, 5901, 6561, 6601, 6721, 6741, 7587

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)) == J(p,D) (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)) == 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=5 and k=2, while U(m) is A001906(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. A001906, A071904, A340097 (a=3, b=1, k=1).

Cf. A340123 (a=5, b=1, k=2), A340124 (a=7, b=1, k=2).

Select[Range[3, 10000, 2], CoprimeQ[#, 5] && CompositeQ[#] &&
Divisible[ ChebyshevU[2*#  - JacobiSymbol[#, 5]  - 1, 3/2] - JacobiSymbol[#, 5],  #] &]
Select[Range[3, 10000, 2], CoprimeQ[#, 5] && CompositeQ[#]
&& Divisible[Fibonacci[2*(2*#-JacobiSymbol[#, 5]), 1] - JacobiSymbol[#, 5], #] &]

A340239 Odd composite integers m such that A001906(3m-J(m,5)) == 3J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

Original entry on oeis.org

9, 49, 63, 141, 161, 207, 323, 341, 377, 441, 671, 901, 1007, 1127, 1281, 1449, 1853, 1891, 2071, 2303, 2407, 2501, 2743, 2961, 3827, 4181, 4623, 5473, 5611, 5777, 6119, 6593, 6601, 6721, 7161, 7567, 8149, 8473, 8961, 9729, 9881

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*J(p,D) (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)*J(m,D) (mod m) are called generalized Lucas pseudoprimes of level k+ and parameter a.
Here b=1, a=3, D=5 and k=3, while U(m) is A001906(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. A001906, A071904, A340097 (a=3, b=1, k=1), A340122 (a=3, b=1, k=2).

Cf. A340240 (a=5, b=1, k=3), A340241 (a=7, b=1, k=3).

Select[Range[3, 10000, 2], CoprimeQ[#, 5] && CompositeQ[#] &&  Divisible[ ChebyshevU[3*#  - JacobiSymbol[#, 5]  - 1, 3/2] - 3*JacobiSymbol[#, 5],  #] &]

cp's OEIS Frontend

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.

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

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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.

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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.

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A340122 Odd composite integers m such that A001906(2*m-J(m,5)) == J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

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A340239 Odd composite integers m such that A001906(3*m-J(m,5)) == 3*J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

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A340239 Odd composite integers m such that A001906(3m-J(m,5)) == 3J(m,5) (mod m), where J(m,5) is the Jacobi symbol.