A339725 -id:A339725 - cp's OEIS frontend

A339724 Odd composite integers m such that A000032(3m-J(m,5)) == 3J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

9, 21, 161, 341, 901, 1281, 1853, 3201, 4181, 5473, 5611, 5777, 6119, 6721, 9729, 10877, 11041, 12209, 12441, 13201, 14981, 15251, 16771, 17941, 20591, 20769, 20801, 23323, 25761, 27403, 27661, 28121, 28421, 29489, 33001, 34561, 38801, 39281, 41159, 42721

Offset: 1

Ovidiu Bagdasar, Dec 14 2020

nonn

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 V(k*p-J(p,D)) == V(k-1)*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 V(k*m-J(m,D)) == V(k-1)*J(m,D) (mod m) are called generalized Pell-Lucas pseudoprimes of level k- and parameter a.
Here b=-1, a=1, D=5 and k=3, while V(m) is A000032(m) (Lucas numbers), with V(2)=3.

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, 24(1), 9-15 (2018).

Cf. A000032, A071904, A339125 (a=1, b=-1, k=1), A339517 (a=1, b=-1, k=2).

Cf. A339725 (a=3, b=-1), A339726 (a=5, b=-1), A339727 (a=7, b=-1).

Select[Range[3, 43000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[LucasL[3*# - JacobiSymbol[#, 5]] - 3*JacobiSymbol[#, 5], #] &]

A339726 Odd composite integers m such that A087130(3m-J(m,29)) == 27J(m,29) (mod m), where J(m,29) is the Jacobi symbol.

Original entry on oeis.org

9, 25, 27, 33, 35, 45, 65, 81, 99, 117, 121, 161, 175, 221, 225, 297, 325, 363, 585, 645, 705, 825, 891, 1089, 1281, 1539, 1541, 1881, 2025, 2133, 2145, 2181, 2299, 2325, 2925, 3025, 3267, 3605, 3745, 4181, 4573, 4579, 5265, 5633, 6721, 6993, 7245, 7425, 7865

Offset: 1

Ovidiu Bagdasar, Dec 14 2020

nonn

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 V(k*p-J(p,D)) == V(k-1)*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 V(k*m-J(m,D)) == V(k-1)*J(m,D) (mod m) are called generalized Pell-Lucas pseudoprimes of level k- and parameter a.
Here b=-1, a=5, D=29 and k=3, while V(m) recovers A087130(m), with V(2)=27.

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, 24(1), 9-15 (2018).

Cf. A087130, A071904, A339127 (a=5, b=-1, k=1), A339519 (a=5, b=-1, k=2).

Cf. A339724 (a=1, b=-1), A339725 (a=3, b=-1), A339727 (a=7, b=-1).

Select[Range[3, 8000, 2],  CoprimeQ[#, 29] && CompositeQ[#] && Divisible[LucasL[3*# - JacobiSymbol[#, 29], 5] - 27*JacobiSymbol[#, 29], #] &]

A339727 Odd composite integers m such that A086902(3m-J(m,53)) == 51J(m,53) (mod m), where J(m,53) is the Jacobi symbol.

Original entry on oeis.org

9, 25, 49, 51, 69, 91, 105, 143, 145, 153, 185, 221, 225, 325, 339, 391, 425, 441, 481, 637, 645, 705, 805, 833, 897, 925, 1001, 1173, 1189, 1207, 1225, 1281, 1299, 1365, 1541, 1633, 1653, 1785, 1813, 1921, 2325, 2599, 2651, 2769, 3133, 3333, 3381, 3605, 3825, 3897

Offset: 1

Ovidiu Bagdasar, Dec 14 2020

nonn

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 V(k*p-J(p,D)) == V(k-1)*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 V(k*m-J(m,D)) == V(k-1)*J(m,D) (mod m) are called generalized Pell-Lucas pseudoprimes of level k- and parameter a.
Here b=-1, a=7, D=53 and k=3, while V(m) recovers A086902(m), with V(2)=51.

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, 24(1), 9-15 (2018).

Cf. A086902, A071904, A339128 (a=7, b=-1, k=1), A339520 (a=7, b=-1, k=2).

Cf. A339724 (a=1, b=-1), A339725 (a=3, b=-1), A339726 (a=5, b=-1).

Select[Range[3, 4000, 2], CoprimeQ[#, 53] && CompositeQ[#] && Divisible[LucasL[3*# - JacobiSymbol[#, 53], 7] - 51*JacobiSymbol[#, 53], #] &]

cp's OEIS Frontend

A339724 Odd composite integers m such that A000032(3m-J(m,5)) == 3J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

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A339726 Odd composite integers m such that A087130(3m-J(m,29)) == 27J(m,29) (mod m), where J(m,29) is the Jacobi symbol.

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A339727 Odd composite integers m such that A086902(3m-J(m,53)) == 51J(m,53) (mod m), where J(m,53) is the Jacobi symbol.

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cp's OEIS Frontend

A339724 Odd composite integers m such that A000032(3*m-J(m,5)) == 3*J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

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A339726 Odd composite integers m such that A087130(3*m-J(m,29)) == 27*J(m,29) (mod m), where J(m,29) is the Jacobi symbol.

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A339727 Odd composite integers m such that A086902(3*m-J(m,53)) == 51*J(m,53) (mod m), where J(m,53) is the Jacobi symbol.

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Mathematica

A339724 Odd composite integers m such that A000032(3m-J(m,5)) == 3J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

A339726 Odd composite integers m such that A087130(3m-J(m,29)) == 27J(m,29) (mod m), where J(m,29) is the Jacobi symbol.

A339727 Odd composite integers m such that A086902(3m-J(m,53)) == 51J(m,53) (mod m), where J(m,53) is the Jacobi symbol.