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.

A337234 Odd composite integers m such that A006190(m)^2 == 1 (mod m).

Original entry on oeis.org

9, 33, 55, 63, 99, 119, 153, 231, 385, 399, 561, 649, 935, 981, 1023, 1071, 1179, 1189, 1199, 1441, 1595, 1763, 1881, 1953, 2001, 2065, 2255, 2289, 2465, 2703, 2751, 2849, 2871, 3519, 3599, 3655, 3927, 4059, 4081, 4187, 5015, 5151, 5559, 6061, 6119, 6215, 6273, 6431
Offset: 1

Views

Author

Ovidiu Bagdasar, Aug 20 2020

Keywords

Comments

If p is a prime, then A006190(p)^2 == 1 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
The generalized Lucas sequence of integer parameters (a,b) defined by U(n+2) = a*U(n+1)-b*U(n) and U(0)=0, U(1)=1, satisfies the identity U^2(p) == 1 (mod p) whenever p is prime and b=-1.
For a=3, b=-1, U(n) recovers A006190(n) ("Bronze" Fibonacci numbers).

References

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

Links

Crossrefs

Cf. A337231 (a=1, odd terms), A337232 (a=1, even terms), A337233 (a=2).

Programs

  • Mathematica
    Select[Range[3, 25000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 3]*Fibonacci[#, 3] - 1, #] &]

A337235 Even composite integers m such that A006190(m)^2 == 1 (mod m).

Original entry on oeis.org

4, 8, 16, 68, 1208, 1424, 3056, 3824, 3928, 20912, 52174, 63716, 88708, 123148, 161872, 582224, 887566, 17083292, 18900412, 34648888, 39991684, 44884912, 51390736, 103170448, 107825236, 132238514, 279900272, 686071244, 769252508, 3251623346, 3358311986, 3535011826
Offset: 1

Views

Author

Ovidiu Bagdasar, Aug 20 2020

Keywords

Comments

If p is a prime, then A006190(p)^2 == 1 (mod p).
This sequence contains the even composite integers for which the congruence holds.
The generalized Lucas sequence of integer parameters (a,b) defined by U(n+2) = a*U(n+1)-b*U(n) and U(0)=0, U(1)=1, satisfies the identity U^2(p) == 1 (mod p) whenever p is prime and b=-1,1.
For a=3, b=-1, U(n) recovers A006190(n) ("Bronze" Fibonacci numbers).

References

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

Links

Crossrefs

Cf. A337231 (a=1, odd terms), A337232 (a=1, even terms), A337233 (a=2), A337234 (a=3, odd terms).

Programs

  • Mathematica
    Select[Range[3, 25000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 5]*Fibonacci[#, 5] - 1, #] &]

Extensions

More terms from Amiram Eldar, Aug 21 2020
a(18)-a(32) from Daniel Suteu, Aug 29 2020

A337236 Composite integers m such that A001076(m)^2 == 1 (mod m).

Original entry on oeis.org

9, 63, 99, 119, 161, 207, 209, 231, 279, 323, 341, 377, 391, 549, 589, 671, 759, 779, 799, 897, 901, 1007, 1159, 1281, 1443, 1449, 1551, 1853, 1891, 2001, 2047, 2071, 2379, 2407, 2501, 2737, 2743, 2849, 2871, 2961, 3069, 3289, 3689, 3827, 4059, 4181, 4199, 4209, 4577
Offset: 1

Views

Author

Ovidiu Bagdasar, Aug 20 2020

Keywords

Comments

If p is a prime, then A001076(p)^2==1 (mod p).
This sequence contains the composite integers for which the congruence holds.
The generalized Lucas sequence of integer parameters (a,b) defined by U(n+2)=a*U(n+1)-b*U(n) and U(0)=0, U(1)=1, satisfies the identity U^2(p)==1 (mod p) whenever p is prime and b=-1,1.
For a=4, b=-1, U(n) recovers A001076(n).

References

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

Links

Crossrefs

Cf. A337231 (a=1, odd terms), A337232 (a=1, even terms), A337233 (a=2), A337234 (a=3, odd terms), A337235 (a=3, even terms).

Programs

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

A337237 Odd composite integers such that A052918(m-1)^2 == 1 (mod m).

Original entry on oeis.org

9, 15, 25, 27, 35, 45, 65, 75, 91, 121, 135, 143, 175, 225, 275, 325, 385, 455, 533, 595, 615, 675, 935, 1035, 1107, 1325, 1359, 1431, 1495, 1547, 1573, 1935, 2015, 2255, 2275, 2775, 3025, 3059, 3575, 3605, 4025, 4081, 4235, 4355, 5005, 5089, 5475, 5525, 5719, 5993, 6165
Offset: 1

Views

Author

Ovidiu Bagdasar, Aug 20 2020

Keywords

Comments

If p is a prime, then A052918(p-1)^2 == 1 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
The generalized Lucas sequence of integer parameters (a,b) defined by U(n+2) = a*U(n+1)-b*U(n) and U(0)=0, U(1)=1, satisfies the identity U^2(p) == 1 (mod p) whenever p is prime and b=-1,1 (this property is a form of pseudoprimality).
For a=5, b=-1, U(n) recovers A052918(n-1), for n=1,2,....

References

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

Links

Crossrefs

Cf. A337231 (a=1, odd terms), A337232 (a=1, even terms), A337233 (a=2), A337234 (a=3, odd terms), A337235 (a=3, even terms), A337236 (a=4).

Programs

  • Mathematica
    Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 5]*Fibonacci[#, 5] - 1, #] &]

A338081 Odd composite integers such that A054413(m)^2 == 1 (mod m).

Original entry on oeis.org

21, 25, 35, 49, 51, 65, 85, 91, 119, 147, 161, 175, 221, 231, 245, 325, 357, 377, 391, 399, 425, 455, 539, 559, 561, 575, 595, 629, 637, 759, 791, 833, 1001, 1105, 1127, 1225, 1247, 1295, 1309, 1495, 1547, 1633, 1763, 1775, 1921, 2001, 2015, 2261, 2275, 2407
Offset: 1

Views

Author

Ovidiu Bagdasar, Oct 08 2020

Keywords

Comments

The generalized Lucas sequence of integer parameters (a,b) is defined by
U(m+2) = a*U(m+1)-b*U(m) and U(0)=0, U(1)=1.
Whenever p is prime and b=-1,1 we have U^2(p) == 1 (mod p).
Here we define the odd composite integers for which U^2(m) == 1 (mod m) holds, for a=7, b=-1, where U(m) is A054413(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)

Links

Crossrefs

Cf. A337231 (a=1, odd terms), A337232 (a=1, even terms), A337233 (a=2), A337234 (a=3, odd terms), A337235 (a=3, even terms), A337236 (a=4), A337237 (a=5), A338081 (a=6).

Programs

  • Mathematica
    Select[Range[3, 15000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 7]*Fibonacci[#, 7] - 1, #] &]

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 *)

A338311 Even composites m such that A003499(m)==6 (mod m).

Original entry on oeis.org

4, 14, 28, 164, 434, 574, 1106, 5084, 5572, 7874, 8386, 13454, 13694, 19964, 21988, 33166, 39934, 40132, 95122, 103886, 113918, 148994, 157604, 215326, 216124, 256004, 277564, 306404, 341342, 366148, 571154, 660674, 662494, 764956, 771374, 876644, 981646, 1070926
Offset: 1

Views

Author

Ovidiu Bagdasar, Oct 22 2020

Keywords

Comments

If p is a prime, then A003499(p)==6 (mod p).
This sequence contains the even 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 and b=1, V(m) recovers A003499(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. A337233 (sequence of odd terms), A337777 (a=3).

Programs

  • Mathematica
    Select[Range[2, 25000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 3] - 6, #] &]

Extensions

More terms from Amiram Eldar, Oct 22 2020

A338080 Odd composite integers such that A005668(m)^2 == 1 (mod m).

Original entry on oeis.org

9, 57, 63, 143, 171, 247, 323, 399, 407, 481, 629, 703, 721, 779, 899, 927, 1121, 1239, 1407, 1441, 1463, 1703, 1729, 2419, 2529, 2639, 2737, 3289, 3367, 3689, 4081, 4847, 4879, 4921, 5291, 5339, 5871, 6061, 6479, 6489, 6601, 6721, 6989, 7067, 7471, 7859, 8401, 8911, 8987, 9139, 9361
Offset: 1

Views

Author

Ovidiu Bagdasar, Oct 08 2020

Keywords

Comments

The generalized Lucas sequence of integer parameters (a,b) is defined by
U(m+2) = a*U(m+1)-b*U(m) and U(0)=0, U(1)=1.
Whenever p is prime and b=-1,1 we have U^2(p) == 1 (mod p).
Here we define the odd composite integers for which U^2(m) == 1 (mod m) holds, for a=6, b=-1, where U(m) is A005668(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. A337231 (a=1, odd terms), A337232 (a=1, even terms), A337233 (a=2), A337234 (a=3, odd terms), A337235 (a=3, even terms), A337236 (a=4), A337237 (a=5).

Programs

  • Mathematica
    Select[Range[3, 15000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 6]*Fibonacci[#, 6] - 1, #] &]
Showing 1-8 of 8 results.

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