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 1-6 of 6 results.

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.

Original entry on oeis.org

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

Views

Author

Ovidiu Bagdasar, Dec 28 2020

Keywords

Comments

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

References

  • D. Andrica and O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.

Links

Crossrefs

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).

Programs

  • Mathematica
    Select[Range[3,28000, 2], CoprimeQ[#, 29] && CompositeQ[#] && Divisible[Fibonacci[#-JacobiSymbol[#, 29], 5], #] &]

Extensions

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

Views

Author

Ovidiu Bagdasar, Dec 28 2020

Keywords

Comments

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

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. 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).

Programs

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

A340097 Odd composite integers m such that A001906(m-J(m,5)) == 0 (mod m) and gcd(m,5)=1, where J(m,5) is the Jacobi symbol.

Original entry on oeis.org

21, 323, 329, 377, 451, 861, 1081, 1819, 1891, 2033, 2211, 3653, 3827, 4089, 4181, 5671, 5777, 6601, 6721, 8149, 8557, 10877, 11309, 11663, 13201, 13861, 13981, 14701, 15251, 17119, 17513, 17711, 17941, 18407, 19043, 19951, 20473, 23407, 25369, 25651, 25877, 27323, 27511
Offset: 1

Views

Author

Ovidiu Bagdasar, Dec 28 2020

Keywords

Comments

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=3 and b=1, we have D=5 and U(m) recovers A001906(m).

References

  • D. Andrica and O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.

Links

Crossrefs

Cf. A001906, A071904, A081264 (a=1, b=-1), A327653 (a=3,b=-1), A340095 (a=5, b=-1), A340096 (a=7, b=-1), A340098 (a=5, b=1), A340099 (a=7, b=1).

Programs

  • Mathematica
    Select[Range[3, 30000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[ChebyshevU[# - JacobiSymbol[#, 5] - 1, 3/2], #] &]

Extensions

Coprime condition added to definition by Georg Fischer, Jul 20 2022

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

Views

Author

Ovidiu Bagdasar, Dec 28 2020

Keywords

Comments

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).

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. 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).

Programs

  • Mathematica
    Select[Range[3, 25000, 2], CoprimeQ[#, 21] && CompositeQ[#] && Divisible[ChebyshevU[# - JacobiSymbol[#, 21] - 1, 5/2], #] &]

A340124 Odd composite integers m such that A004187(2*m-J(m,45)) == J(m,45) (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol.

Original entry on oeis.org

49, 323, 329, 343, 377, 451, 1081, 1127, 1771, 1819, 1891, 2033, 2303, 2401, 3653, 3827, 4181, 5671, 5777, 6601, 6721, 7471, 7931, 8149, 8557, 9691, 10877, 11309, 11663, 13201, 13861, 13981, 14701, 15251, 15449, 16121, 16807, 17119, 17513, 17687, 17711
Offset: 1

Views

Author

Ovidiu Bagdasar, Dec 28 2020

Keywords

Comments

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=7, D=45 and k=2, while U(m) is A004187(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. A004187, A071904, A340099 (a=7, b=1, k=1).
Cf. A340122 (a=3, b=1, k=2), A340123 (a=5, b=1, k=2).

Programs

  • Mathematica
    Select[Range[3, 20000, 2], CoprimeQ[#, 45] && CompositeQ[#] &&
Divisible[ ChebyshevU[2*# - JacobiSymbol[#, 45] - 1, 7/2] - JacobiSymbol[#, 45],  #] &]

A340241 Odd composite integers m such that A004187(3*m-J(m,45)) == 7*J(m,45) (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol.

Original entry on oeis.org

161, 323, 329, 341, 377, 451, 671, 901, 1007, 1079, 1081, 1271, 1819, 1853, 1891, 2033, 2071, 2209, 2407, 2461, 2501, 2743, 3653, 3827, 4181, 4843, 5473, 5611, 5671, 5777, 6119, 6601, 6721, 7429, 7567, 7721, 8149, 8399, 8473, 8557, 9821, 9881, 10207, 10877, 11041, 11207, 11309, 11663
Offset: 1

Views

Author

Ovidiu Bagdasar, Jan 01 2021

Keywords

Comments

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=7, D=45 and k=3, while U(m) is A004187(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. A004187, A071904, A340099 (a=7, b=1, k=1), A340124 (a=7, b=1, k=2).
Cf. A340239 (a=3, b=1, k=3), A340240 (a=5, b=1, k=3).

Programs

  • Mathematica
    Select[Range[3, 12000, 2], CoprimeQ[#, 45] && CompositeQ[#] &&  Divisible[ ChebyshevU[3*# - JacobiSymbol[#, 45] - 1, 7/2] - 7*JacobiSymbol[#, 45],  #] &]
Showing 1-6 of 6 results.

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