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 1-3 of 3 results.

A335673 Composite integers m such that A003500(m) == 4 (mod m).

Original entry on oeis.org

10, 209, 230, 231, 399, 430, 455, 530, 901, 903, 923, 989, 1295, 1729, 1855, 2015, 2211, 2345, 2639, 2701, 2795, 2911, 3007, 3201, 3439, 3535, 3801, 4823, 5291, 5719, 6061, 6767, 6989, 7421, 8569, 9503, 9591, 9869, 9890, 10439, 10609, 11041, 11395, 11951, 11991
Offset: 1

Views

Author

Ovidiu Bagdasar, Jun 17 2020

Keywords

Comments

If p is a prime, then A003500(p)==4 (mod p).
This sequence contains the composite integers for which the congruence holds.
The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a, satisfy the identity V(p)==a (mod p) whenever p is prime and b=-1,1.
For a=4, b=1, V(n)=A003500(n).

Examples

			m=10 is the first composite integer for which A003500(m)==4 (mod m).

References

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

Links

Crossrefs

Cf. A005248, A335669 (a=3,b=-1), A335672 (a=3,b=1), A335674 (a=5,b=1).
A330206 is the subsequence of odd terms.

Programs

  • Mathematica
    Select[Range[3, 20000], CompositeQ[#] && Divisible[Round@LucasL[2#, Sqrt[2]] - 4, #] &] (* Amiram Eldar, Jun 18 2020 *)
  • PARI
    my(M=[1,2;1,3]); forcomposite(m=5, 10^5, if(trace(Mod(M,m)^m)==4, print1(m,", "))); \\ Joerg Arndt, Jun 18 2020

Extensions

More terms from Joerg Arndt, Jun 18 2020

A335674 Odd composite integers m such that A003501(m) == 5 (mod m).

Original entry on oeis.org

15, 21, 35, 105, 161, 195, 255, 345, 385, 399, 465, 527, 551, 609, 741, 897, 1105, 1295, 1311, 1807, 1919, 2001, 2015, 2071, 2085, 2121, 2415, 2737, 2915, 3289, 3815, 4031, 4033, 4355, 4879, 4991, 5291, 5777, 5983, 6049, 6061, 6083, 6479, 6601, 6785, 7645, 7905, 8695, 8855, 8911, 9361, 9591, 9889
Offset: 1

Views

Author

Ovidiu Bagdasar, Jun 17 2020

Keywords

Comments

If p is a prime, then A003501(p)==5 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a, satisfy the identity V(p)==a (mod p) whenever p is prime and b=-1,1.
For a=5, b=1, V(n) recovers A003501(n).

Examples

			15 is the first odd composite integer for which the relation A003501(15)=16098445550==5 (mod 15) holds.

References

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

Links

Crossrefs

Cf. A005248, A335669 (a=3,b=-1), A335672 (a=3,b=1), A335673 (a=4,b=1).

Programs

  • Mathematica
    Select[Range[3, 5000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 5/2] - 5, #] &] (* Amiram Eldar, Jun 18 2020 *)

A338082 Odd composite integers m such that A056854(m) == 7 (mod m).

Original entry on oeis.org

9, 15, 21, 35, 45, 63, 99, 105, 195, 231, 315, 323, 329, 369, 377, 423, 435, 451, 595, 665, 705, 805, 861, 903, 1081, 1189, 1443, 1551, 1819, 1833, 1869, 1891, 1935, 2033, 2211, 2345, 2465, 2737, 2849, 2871, 2961, 3059, 3289, 3653, 3689, 3745, 3827, 4005, 4059
Offset: 1

Views

Author

Ovidiu Bagdasar, Oct 08 2020

Keywords

Comments

If p is a prime, then A056854(p)==7 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
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)==a (mod p) when p is prime and b=-1,1.
For a=7 and b=1, 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)

Crossrefs

Cf. A005248, A335669 (a=3,b=-1), A335672 (a=3,b=1), A335673 (a=4,b=1), A335674 (a=5,b=1), A337233 (a=6,b=1).

Programs

  • Mathematica
    Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 7/2] - 7, #] &]
Select[Range[9,5001,2],CompositeQ[#]&&Mod[LucasL[4#],#]==7&] (* Harvey P. Dale, Apr 28 2022 *)
Showing 1-3 of 3 results.

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