A340235 -id:A340235 - cp's OEIS frontend

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

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, #] &]

A340237 Odd composite integers m such that A052918(3*m-J(m,29)) == 5 (mod m), where J(m,29) is the Jacobi symbol.

Original entry on oeis.org

9, 27, 33, 35, 65, 81, 99, 121, 221, 243, 297, 363, 513, 585, 627, 705, 729, 891, 1089, 1539, 1541, 1881, 2145, 2187, 2299, 2673, 3267, 3605, 4181, 4573, 4579, 5265, 5633, 6721, 6993, 7865, 8019, 8979, 9131, 9801, 10307, 10877, 10881, 13333, 13741, 14001, 14705, 14989

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=5, D=29 and k=3, while U(m) is A052918(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. A052918, A071904, A340095 (a=5, b=-1, k=1), A340120 (a=5, b=-1, k=2).

Cf. A340235 (a=1, b=-1, k=3), A340236 (a=3, b=-1, k=3), A340238 (a=7, b=-1, k=3).

Select[Range[3, 15000, 2], CoprimeQ[#, 29] && CompositeQ[#] && Divisible[Fibonacci[3*#-JacobiSymbol[#, 29], 5] - 5, #] &]

A340238 Odd composite integers m such that A054413(3*m-J(m,53)) == 7 (mod m), where J(m,53) is the Jacobi symbol.

Original entry on oeis.org

9, 25, 27, 51, 91, 105, 153, 185, 225, 289, 325, 425, 459, 481, 513, 747, 867, 897, 925, 945, 1001, 1189, 1299, 1469, 1633, 1785, 1921, 2241, 2245, 2599, 2601, 2651, 2769, 2907, 3051, 3277, 3825, 3897, 5681, 6225, 6507, 6777, 7225, 7361, 7803, 8023, 8227, 8701, 8721

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=7, D=53 and k=3, while U(m) is A054413(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. A054413, A071904, A340096 (a=7, b=-1, k=1), A340121 (a=7, b=-1, k=2).

Cf. A340235 (a=1, b=-1, k=3), A340236 (a=3, b=-1, k=3), A340237 (a=5, b=-1, k=3).

Select[Range[3, 10000, 2], CoprimeQ[#, 53] && CompositeQ[#] && Divisible[Fibonacci[3*#-JacobiSymbol[#, 53], 7] - 7, #] &]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

A340237 Odd composite integers m such that A052918(3*m-J(m,29)) == 5 (mod m), where J(m,29) is the Jacobi symbol.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

A340238 Odd composite integers m such that A054413(3*m-J(m,53)) == 7 (mod m), where J(m,53) is the Jacobi symbol.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica