cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A339125 Odd composite integers m such that A000032(m-J(m,5)) == 2*J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

Original entry on oeis.org

9, 49, 121, 169, 289, 361, 529, 841, 961, 1127, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 3751, 4181, 4489, 4901, 4961, 5041, 5329, 5777, 6241, 6721, 6889, 7381, 7921, 9409, 10201, 10609, 10877, 11449, 11881, 12769, 13201, 15251, 16129, 17161, 18081, 18769, 19321
Offset: 1

Views

Author

Ovidiu Bagdasar, Nov 24 2020

Keywords

Comments

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-J(p,D)) == 2*J(p,D) (mod p) when p is prime, b=-1 and D=a^2+4.
This sequence has the odd composite integers with V(m-J(m,D)) == 2*J(m,D) (mod m).
For a=1 and b=-1, we have D=5 and V(m) recovers A000032(m) (Lucas numbers).

References

  • 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)

Links

Crossrefs

Cf. A339126 (a=3, b=-1), A339127 (a=5, b=-1), A339128 (a=7, b=-1), A339129 (a=3, b=1), A339130 (a=5, b=1), A339131 (a=7, b=1).

Programs

  • Mathematica
    Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[LucasL[# - (j = JacobiSymbol[#, 5])] - 2*j, #] &] (* Amiram Eldar, Nov 26 2020 *)

A339126 Odd composite integers m such that A006497(m-J(m,13)) == 2*J(m,13) (mod m), where J(m,13) is the Jacobi symbol.

Original entry on oeis.org

9, 25, 49, 119, 121, 289, 361, 529, 649, 833, 841, 961, 1089, 1189, 1369, 1681, 1849, 1881, 2023, 2209, 2299, 2809, 3025, 3481, 3721, 4187, 4489, 5041, 5329, 6241, 6889, 7139, 7921, 9409, 10201, 10241, 10609, 11449, 11881, 12769, 12871, 13833, 14041, 14161
Offset: 1

Views

Author

Ovidiu Bagdasar, Nov 24 2020

Keywords

Comments

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-J(p,D)) == 2*J(p,D) (mod p) when p is prime, b=-1 and D=a^2+4.
This sequence has the odd composite integers with V(m-J(m,D)) == 2*J(m,D) (mod m).
For a=3 and b=-1, we have D=13 and V(m) recovers A006497(m).

References

  • 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)

Crossrefs

Cf. A006497.
Cf. A339125 (a=1, b=-1), A339127 (a=5, b=-1), A339128 (a=7, b=-1), A339129 (a=3, b=1), A339130 (a=5, b=1), A339131 (a=7, b=1).

Programs

  • Mathematica
    Select[Range[3, 15000, 2], CompositeQ[#] && Divisible[LucasL[# - (j = JacobiSymbol[#, 13]), 3] - 2*j, #] &] (* Amiram Eldar, Nov 26 2020 *)

A339127 Odd composite integers m such that A087130(m-J(m,29)) == 2*J(m,29) (mod m), where J(m,29) is the Jacobi symbol.

Original entry on oeis.org

9, 25, 27, 49, 81, 121, 169, 175, 225, 243, 289, 325, 361, 529, 637, 729, 961, 1225, 1331, 1369, 1539, 1681, 1849, 2025, 2209, 2809, 3025, 3481, 3721, 4225, 4489, 5041, 5329, 5929, 6241, 6721, 6889, 6929, 7921, 8281, 9409
Offset: 1

Views

Author

Ovidiu Bagdasar, Nov 24 2020

Keywords

Comments

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-J(p,D)) == 2*J(p,D) (mod p) when p is prime, b=-1 and D=a^2+4.
This sequence has the odd composite integers with V(m-J(m,D)) == 2*J(m,D) (mod m).
For a=5 and b=-1, we have D=29 and V(m) recovers A087130(m).

References

  • 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)

Crossrefs

Cf. A087130.
Cf. A339125 (a=1, b=-1), A339126 (a=3, b=-1), A339128 (a=7, b=-1), A339129 (a=3, b=1), A339130 (a=5, b=1), A339131 (a=7, b=1).

Programs

  • Mathematica
    Select[Range[3, 10000, 2], CompositeQ[#] && Divisible[LucasL[# - (j = JacobiSymbol[#, 29]), 5] - 2*j, #] &] (* Amiram Eldar, Nov 26 2020 *)

A339129 Odd composite integers m such that A005248(m-J(m,5)) == 2 (mod m), where J(m,5) is the Jacobi symbol.

Original entry on oeis.org

9, 49, 63, 121, 169, 289, 323, 361, 377, 441, 529, 841, 961, 1127, 1369, 1681, 1849, 1891, 2209, 2303, 2809, 2961, 3481, 3721, 3751, 3827, 4181, 4489, 4901, 4961, 5041, 5329, 5491, 5777, 6137, 6241, 6601, 6721, 6889, 7381, 7921, 8149, 9409, 10201, 10609, 10877, 10933, 11449, 11663
Offset: 1

Views

Author

Ovidiu Bagdasar, Nov 24 2020

Keywords

Comments

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-J(p,D)) == 2 (mod p) when p is prime, b=1 and D=a^2-4.
This sequence contains the odd composite integers with V(m-J(m,D)) == 2 (mod m).
For a=3 and b=1, we have D=5 and V(m) recovers A005248(m).

References

  • 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)

Crossrefs

Cf. A005248.
Cf. A339125 (a=1, b=-1), A339126 (a=3, b=-1), A339127 (a=5, b=-1), A339128 (a=7, b=-1), A339130 (a=5, b=1), A339131 (a=7, b=1).

Programs

  • Mathematica
    Select[Range[3, 12000, 2], CompositeQ[#] && Divisible[LucasL[2*(# - JacobiSymbol[#, 5])] - 2, #] &] (* Amiram Eldar, Nov 26 2020 *)

A339130 Odd composite integers m such that A003501(m-J(m,21)) == 2 (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.

Original entry on oeis.org

25, 121, 169, 275, 289, 361, 527, 529, 551, 575, 841, 961, 1369, 1681, 1807, 1849, 1919, 2209, 2783, 2809, 3025, 3481, 3721, 4033, 4489, 5041, 5329, 5777, 5983, 6049, 6241, 6479, 6575, 6889, 7267, 7645, 7921, 8959, 8993, 9361, 9409, 9775
Offset: 1

Views

Author

Ovidiu Bagdasar, Nov 24 2020

Keywords

Comments

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-J(p,D)) == 2 (mod p) when p is prime, b=1 and D=a^2-4.
This sequence contains the odd composite integers with V(m-J(m,D)) == 2 (mod m).
For a=5 and b=1, we have D=21 and V(m) recovers A003501(m).

References

  • 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)

Crossrefs

Cf. A003501.
Cf. A339125 (a=1, b=-1), A339126 (a=3, b=-1), A339127 (a=5, b=-1), A339128 (a=7, b=-1), A339129 (a=3, b=1), A339131 (a=7, b=1).

Programs

  • Mathematica
    Select[Range[3, 10000, 2], CoprimeQ[#, 21] && CompositeQ[#] && Divisible[2*ChebyshevT[# - JacobiSymbol[#, 21], 5/2] - 2, #] &] (* Amiram Eldar, Nov 26 2020 *)

A339131 Odd composite integers m such that A056854(m-J(m,45)) == 2 (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol.

Original entry on oeis.org

49, 121, 169, 289, 323, 329, 361, 377, 451, 529, 841, 961, 1081, 1127, 1369, 1681, 1819, 1849, 1891, 2033, 2209, 2303, 2809, 3481, 3653, 3721, 3751, 3827, 4181, 4489, 4901, 4961, 5041, 5329, 5491, 5671, 5777, 6137, 6241, 6601, 6721, 6889, 7381, 7921, 8149, 8557, 9409
Offset: 1

Views

Author

Ovidiu Bagdasar, Nov 24 2020

Keywords

Comments

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-J(p,D)) == 2 (mod p) when p is prime, b=1 and D=a^2-4.
This sequence contains the odd composite integers with V(m-J(m,D)) == 2 (mod m).
For a=7 and b=1, we have D=45 and V(m) recovers A056854(m).

References

  • 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)

Crossrefs

Cf. A056854.
Cf. A339125 (a=1, b=-1), A339126 (a=3, b=-1), A339127 (a=5, b=-1), A339128 (a=7, b=-1), A339129 (a=3, b=1), A339130 (a=5, b=1).

Programs

  • Mathematica
    Select[Range[3, 10000, 2], CoprimeQ[#, 45] && CompositeQ[#] && Divisible[LucasL[4*(# - JacobiSymbol[#, 45])] - 2, #] &] (* Amiram Eldar, Nov 26 2020 *)

A339520 Odd composite integers m such that A086902(2*m-J(m,53)) == 7*J(m,53) (mod m), where J(m,53) is the Jacobi symbol.

Original entry on oeis.org

25, 35, 51, 65, 75, 91, 105, 175, 203, 325, 391, 455, 575, 645, 861, 1247, 1275, 1295, 1633, 1763, 1775, 1785, 1875, 1921, 2275, 2407, 2415, 2599, 2625, 2651, 3045, 3367, 4199, 4579, 4623, 5629, 5835, 5887, 6441, 6699, 9959, 10465, 10815, 10825, 10877, 11865, 12025
Offset: 1

Views

Author

Ovidiu Bagdasar, Dec 07 2020

Keywords

Comments

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=2, while V(m) recovers A086902(m).

References

  • 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).

Links

Crossrefs

Cf. A086902, A071904, A339128 (a=7, b=-1, k=1).
Cf. A339517 (a=1, b=-1), A339518 (a=3, b=-1), A339529 (a=5, b=-1).

Programs

  • Mathematica
    Select[Range[3, 20000, 2], CoprimeQ[#, 53] && CompositeQ[#] && Divisible[LucasL[2*# - JacobiSymbol[#, 53], 7] - 7*JacobiSymbol[#, 53], #] &]

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.

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

Views

Author

Ovidiu Bagdasar, Dec 14 2020

Keywords

Comments

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.

References

  • 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).

Links

Crossrefs

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).

Programs

  • Mathematica
    Select[Range[3, 4000, 2], CoprimeQ[#, 53] && CompositeQ[#] && Divisible[LucasL[3*# - JacobiSymbol[#, 53], 7] - 51*JacobiSymbol[#, 53], #] &]
Showing 1-8 of 8 results.

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