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 1-7 of 7 results.

A335669 Odd composite integers m such that A006497(m) == 3 (mod m).

Original entry on oeis.org

33, 65, 119, 273, 377, 385, 533, 561, 649, 1105, 1189, 1441, 2065, 2289, 2465, 2849, 4187, 4641, 6545, 6721, 11921, 12871, 13281, 14041, 15457, 16109, 18241, 19201, 22049, 23479, 24769, 25345, 28421, 30745, 31631, 34997, 38121, 38503, 41441, 45961, 46761, 48577
Offset: 1

Views

Author

Ovidiu Bagdasar, Jun 17 2020

Keywords

Comments

If p is a prime, then A006497(p) == 3 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
The generalized Pell-Lucas sequence 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=3, b=-1, V(n) recovers the sequence A006497(n) (bronze Fibonacci numbers).

Examples

			33 is the first odd composite integer for which we have A006497(33) = 132742316047301964 == 3 (mod 33).

References

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

Links

Crossrefs

Cf. A006497, A005845 (a=1), A330276 (a=2), A335670 (a=4), A335671 (a=5).

Programs

  • Mathematica
    Select[Range[3, 50000, 2], CompositeQ[#] && Divisible[LucasL[#, 3] - 3, #] &] (* Amiram Eldar, Jun 18 2020 *)
  • PARI
    is(m) = m%2 && !isprime(m) && [2, 3]*([0, 1; 1, 3]^m)[, 1]%m==3; \\ Jinyuan Wang, Jun 17 2020

Extensions

More terms from Jinyuan Wang, Jun 17 2020

A335671 Odd composite integers m such that A087130(m) == 5 (mod m).

Original entry on oeis.org

9, 27, 65, 121, 145, 377, 385, 533, 1035, 1189, 1305, 1885, 2233, 2465, 4081, 5089, 5993, 6409, 6721, 7107, 10877, 11281, 11285, 13281, 13369, 13741, 13833, 14705, 15457, 16721, 17545, 18901, 19601, 19951, 20329, 20705, 22881, 24769, 25345, 26599, 26937, 28741, 29161
Offset: 1

Views

Author

Ovidiu Bagdasar, Jun 17 2020

Keywords

Comments

If p is a prime, then A087130(p)==5 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
The generalized Pell-Lucas sequence 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 A087130(n).

Examples

			9 is the first odd composite integer for which A087130(9)=2744420==5 (mod 9).

References

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

Links

Crossrefs

Cf. A006497, A005845 (a=1), A330276 (a=2), A335669 (a=3), A335670 (a=4).

Programs

  • Maple
    M:= <<5|1>,<1|0>>:
f:= proc(n) uses LinearAlgebra:-Modular;
local A;
A:= Mod(n,M,integer[8]);
A:= MatrixPower(n,A,n);
2*A[1,1] - 5*A[1,2] mod n;
end proc:
select(t -> f(t) = 5 and not isprime(t), [seq(i,i=3..10^5,2)]); # Robert Israel, Jun 19 2020
  • Mathematica
    Select[Range[3, 30000, 2], CompositeQ[#] && Divisible[LucasL[#, 5] - 5, #] &] (* Amiram Eldar, Jun 18 2020 *)

Extensions

More terms from Jinyuan Wang, Jun 17 2020

A337627 Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 4 (mod m), where U(m) and V(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=4 and b=-1, respectively.

Original entry on oeis.org

9, 161, 341, 897, 901, 1281, 1853, 2737, 4181, 4209, 4577, 5473, 5611, 5777, 6119, 6721, 9701, 9729, 10877, 11041, 12209, 12349, 13201, 13481, 14981, 15251, 16771, 19669, 20591, 20769, 20801, 23323, 27403, 27613, 28421, 29281, 29489, 32929, 33001, 34561, 38801
Offset: 1

Views

Author

Ovidiu Bagdasar, Sep 19 2020

Keywords

Comments

Intersection of A335670 and A337236.
For a,b integers, the following sequences are defined:
generalized Lucas sequences by U(n+2)=a*U(n+1)-b*U(n) and U(0)=0, U(1)=1,
generalized Pell-Lucas sequences by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a.
These satisfy the identities U(p)^2 == 1 and V(p)==a (mod p) for p prime and b=1,-1.
These numbers may be called weak generalized Lucas-Bruckner pseudoprimes of parameters a and b.The current sequence is defined for a=4 and b=-1.

Links

Crossrefs

Cf. A335670 and A337236. Similar sequences: A337625 (a=1), A337626 (a=3).

Programs

  • Mathematica
    Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 4]*Fibonacci[#, 4] - 1, #] && Divisible[LucasL[#, 4] - 4, #] &]

Extensions

More terms from Amiram Eldar, Sep 19 2020

A335721 Integers m such that A014448(m) == 5 (mod m).

Original entry on oeis.org

1, 213, 85887, 974943, 2463831, 3952791, 217643749, 286354743, 874273639, 14029228621
Offset: 1

Views

Author

Chai Wah Wu, Jun 18 2020

Keywords

Comments

Inspired by A335670.
All terms greater than 1 are odd and composite. - Michel Marcus, Jun 19 2020

Crossrefs

Cf. A014448, A335670, A335722.

Programs

  • Mathematica
    Select[Range[10^5], Divisible[LucasL[3#] - 5, #] &] (* Amiram Eldar, Jun 19 2020 *)
  • PARI
    f(n) = my(w=quadgen(5)); (1+2*w)^n + (3-2*w)^n; \\ A014448
isok(m) = (m%2) && (m>1) && !isprime(m) && ((f(m) % m) == 5); \\ Michel Marcus, Jun 19 2020

Extensions

a(7)-a(10) from Giovanni Resta, Jun 19 2020

A335722 Integers m such that A014448(m) == 1 (mod m).

Original entry on oeis.org

1, 3, 77, 235, 517, 8155, 17567, 18235, 22827, 33355, 57053, 59899, 67947, 107067, 107767, 151987, 232379, 238539, 289155, 306859, 331115, 360267, 411803, 427467, 471115, 576987, 611187, 681963, 713767, 742467, 765195, 811947, 871827, 982315, 1043915, 1174859, 1211115
Offset: 1

Views

Author

Chai Wah Wu, Jun 18 2020

Keywords

Comments

Inspired by A335670.
All terms > 3 are odd and composite. - Michel Marcus, Jun 19 2020

Crossrefs

Cf. A014448, A335670, A335721.

Programs

  • Mathematica
    Select[Range[10^4], Divisible[LucasL[3#] - 1, #] &] (* Amiram Eldar, Jun 19 2020 *)

A338078 Odd composite integers m such that A085447(m) == 6 (mod m).

Original entry on oeis.org

57, 185, 385, 481, 629, 721, 779, 1121, 1441, 1729, 2419, 2737, 5665, 6721, 7471, 8401, 9361, 10465, 10561, 11285, 11521, 11859, 12257, 13585, 14705, 15281, 16321, 16583, 18849, 24721, 25345, 25441, 25593, 30745, 33649, 35219, 36481, 36581, 37949, 38665, 39169
Offset: 1

Views

Author

Ovidiu Bagdasar, Oct 08 2020

Keywords

Comments

If p is a prime, then A085447(p)==6 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
The generalized Pell-Lucas sequence 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) whenever p is prime and b=-1,1.
For a=6, b=-1, V(m) recovers A085447(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. A006497, A005845 (a=1), A330276 (a=2), A335669 (a=3), A335670 (a=4), A335671 (a=5).

Programs

  • Mathematica
    Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[LucasL[#, 6] - 6, #] &]

Extensions

More terms from Amiram Eldar, Oct 09 2020

A338079 Odd composite integers m such that A086902(m) == 7 (mod m).

Original entry on oeis.org

25, 51, 91, 161, 265, 325, 425, 561, 791, 1105, 1113, 1325, 1633, 1921, 1961, 2001, 2465, 2599, 2651, 2737, 3445, 4081, 4505, 4929, 7345, 7685, 8449, 9361, 10325, 10465, 10825, 11285, 11713, 12025, 12291, 13021, 15457, 17111, 18193, 18881, 18921, 19307
Offset: 1

Views

Author

Ovidiu Bagdasar, Oct 08 2020

Keywords

Comments

If p is a prime, then A086902(p)==7 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
The generalized Pell-Lucas sequence 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) whenever p is prime and b=-1,1.
For a=7, b=-1, 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)

Crossrefs

Cf. A006497, A005845 (a=1), A330276 (a=2), A335669 (a=3), A335670 (a=4), A335671 (a=5), A338078 (a=6).

Programs

  • Mathematica
    Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[LucasL[#, 7] - 7, #] &]
Showing 1-7 of 7 results.

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