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

A335670 Odd composite integers m such that A014448(m) == 4 (mod m).

Original entry on oeis.org

9, 85, 161, 341, 705, 897, 901, 1105, 1281, 1853, 2465, 2737, 3745, 4181, 4209, 4577, 5473, 5611, 5777, 6119, 6721, 9701, 9729, 10877, 11041, 12209, 12349, 13201, 13481, 14981, 15251, 16185, 16545, 16771, 19669, 20591, 20769, 20801, 21845, 23323, 24465, 25345
Offset: 1

Views

Author

Ovidiu Bagdasar, Jun 17 2020

Keywords

Comments

If p is a prime, then A014448(p)==4 (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=4, b=-1, V(n) recovers A014448(n) (even Lucas numbers).

Examples

			9 is the first odd composite integer for which A014448(9)=439204==4 (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), A335671 (a=5).

Programs

  • Maple
    M:= <<4|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] - 4*A[1,2] mod n;
end proc:
select(t -> f(t) = 4 and not isprime(t), [seq(i,i=3..10^5,2)]); # Robert Israel, Jun 19 2020
  • Mathematica
    Select[Range[3, 25000, 2], CompositeQ[#] && Divisible[LucasL[3#] - 4, #] &] (* Amiram Eldar, Jun 18 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

A335668 Even composites m such that A002203(m) == 2 (mod m).

Original entry on oeis.org

4, 8, 16, 24, 32, 48, 64, 72, 96, 120, 128, 144, 168, 192, 216, 240, 256, 264, 272, 288, 336, 360, 384, 432, 480, 504, 512, 528, 544, 576, 600, 648, 672, 720, 768, 792, 816, 840, 864, 960, 1008, 1024, 1056, 1080, 1088, 1152, 1176, 1200, 1296, 1320, 1344, 1440, 1512
Offset: 1

Views

Author

Ovidiu Bagdasar, Jun 17 2020

Keywords

Comments

If p is a prime, then A002203(p)==2 (mod p).
Even composites for which the congruence holds.
Even composites m for which the sum of the Pell numbers A000129(0) + ... + A000129(m-1) is divisible by m.

Examples

			4 is the first composite number m for which A002203(m)==2 (mod m) since A002203(4)=34==2 (mod 4), so a(1)=4.
The next even composite for which the congruence holds is m = 8 since A002203(8)=1154==2 (mod 8), so a(2)=8.

References

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

Links

Crossrefs

Cf. A270342 (all positive integers), A270345 (all composites), A330276 (odd composites),

Programs

  • Mathematica
    Select[Range[4, 2000, 2], Divisible[LucasL[#, 2] - 2, #] &] (* Amiram Eldar, Jun 18 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, #] &]

A372946 Numbers k that divide the k-th NSW number.

Original entry on oeis.org

1, 7, 217, 3937, 6727, 6847, 51943, 170671, 330337, 385687, 2484247, 2566537, 2904007, 3020857, 3696967, 6465577, 9405337, 12021439, 19384207
Offset: 1

Views

Author

Amiram Eldar, May 17 2024

Keywords

Comments

Numbers k such that k | A002315(k).

Examples

			7 is a term since A002315(7) = 275807 = 7 * 39401 is divisible by 7.

Crossrefs

Cf. A002315, A330276.
Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

Programs

  • Mathematica
    seq[kmax_] := Module[{nsw0 = 1, nsw1 = 7, nsw2, s = {1}}, Do[nsw2 = 6*nsw1 - nsw0; If[Divisible[nsw2, k], AppendTo[s, k]]; nsw0 = nsw1; nsw1 = nsw2, {k, 2, kmax}]; s]; seq[52000]
  • PARI
    lista(kmax) = {my(nsw0 = 1, nsw1 = 7, nsw2); print1("1, "); for(k = 2, kmax, nsw2 = 6*nsw1 - nsw0; if(!(nsw2 % k), print1(k, ", ")); nsw0 = nsw1; nsw1 = nsw2);}
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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