A337232 -id:A337232 - cp's OEIS frontend

A337233 Composite integers m such that P(m)^2 == 1 (mod m), where P(m) is the m-th Pell number A000129(m). Also, odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 6 (mod m), where U(m)=A001109(m) and V(m)=A003499(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=6 and b=1, respectively.

35, 119, 169, 385, 741, 779, 899, 935, 961, 1105, 1121, 1189, 1443, 1479, 2001, 2419, 2555, 2915, 3059, 3107, 3383, 3605, 3689, 3741, 3781, 3827, 4199, 4795, 4879, 4901, 5719, 6061, 6083, 6215, 6265, 6441, 6479, 6601, 6895, 6929, 6931, 6965, 7055, 7107, 7801, 8119

Offset: 1

Ovidiu Bagdasar, Aug 20 2020

nonn

For a, b integers, the following sequences are defined:
generalized Lucas sequences by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1,
generalized Pell-Lucas sequences by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a.
In general, one has U^2(p) == 1 and V(p)==a (mod p) whenever p is prime and b=1, -1.
The composite numbers satisfying these congruences may be called weak generalized Lucas-Bruckner pseudoprimes of parameters a and b.
For a=2 and b=-1, U(m) recovers A000129(m) (Pell numbers).
For a=6 and b=1, we have U(m)=A001109(m) and V(m)=A003499(m).
This sequence contains the odd composite integers for which the congruence A000129(m)^2 == 1 (mod m) holds.
This is also the sequence of odd composite numbers satisfying the congruences A001109(m)^2 == 1 and A003499(m)==a (mod m).

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

D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, preprint for Mediterr. J. Math. 18, 47 (2021).

Cf. A337231 (a=1, odd terms), A337232 (a=1, even terms), A337629 (a=6, b=-1), A337778 (a=4, b=1), A337779 (a=5, b=1).

Select[Range[3, 25000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 2]*Fibonacci[#, 2] - 1, #] &]
Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 3] - 6, #] && Divisible[ChebyshevU[#-1, 3]*ChebyshevU[#-1, 3] - 1, #] &]

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

Ovidiu Bagdasar, Aug 20 2020

nonn

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).

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

Dorin Andrica and Ovidiu Bagdasar, On Generalized Lucas Pseudoprimality of Level k, Mathematics (2021) Vol. 9, 838.
D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, preprint for Mediterr. J. Math. 18, 47 (2021).

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

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

Ovidiu Bagdasar, Aug 20 2020

nonn

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).

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

D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, preprint for Mediterr. J. Math. 18, 47 (2021).

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

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

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

Ovidiu Bagdasar, Aug 20 2020

nonn

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).

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

D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, preprint for Mediterr. J. Math. 18, 47 (2021).

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

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

Ovidiu Bagdasar, Aug 20 2020

nonn

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,....

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

Amiram Eldar, Table of n, a(n) for n = 1..1000
Dorin Andrica and Ovidiu Bagdasar, On Generalized Lucas Pseudoprimality of Level k, Mathematics (2021) Vol. 9, 838.
D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, preprint for Mediterr. J. Math. 18, 47 (2021).

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).

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

Ovidiu Bagdasar, Oct 08 2020

nonn

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).

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)

Dorin Andrica and Ovidiu Bagdasar, On Generalized Lucas Pseudoprimality of Level k, Mathematics (2021) Vol. 9, 838.

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).

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

A356296 a(n) = Fibonacci(n)^2 mod n.

Original entry on oeis.org

0, 1, 1, 1, 0, 4, 1, 1, 4, 5, 1, 0, 1, 1, 10, 9, 1, 10, 1, 5, 4, 1, 1, 0, 0, 1, 22, 9, 1, 10, 1, 25, 4, 1, 25, 0, 1, 1, 4, 25, 1, 22, 1, 9, 40, 1, 1, 0, 22, 25, 4, 9, 1, 10, 25, 49, 4, 1, 1, 0, 1, 1, 22, 25, 25, 64, 1, 9, 4, 15, 1, 0, 1, 1, 25, 9, 4, 64, 1, 25, 49, 1, 1, 72, 25, 1

Offset: 1

R. J. Mathar, Aug 03 2022

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000045, A002708, A023172 (location of zeros), A337231, A337232.

A356296 := proc(n)
    modp(combinat[fibonacci](n)^2,n) ;
end proc:
seq(A356296(n),n=1..120) ;

Array[PowerMod[Fibonacci[#], 2, #] &, 86] (* Michael De Vlieger, Aug 03 2022 *)

a(n) = lift(Mod(fibonacci(n), n)^2); \\ Michel Marcus, Aug 03 2022

from sympy import fibonacci
def a(n): return pow(fibonacci(n), 2, n)
print([a(n) for n in range(1, 87)]) # Michael S. Branicky, Aug 04 2022

a(n) = A000045(n)^2 mod n.

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

Ovidiu Bagdasar, Oct 08 2020

nonn

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).

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)

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).

Select[Range[3, 15000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 6]*Fibonacci[#, 6] - 1, #] &]

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

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

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

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

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

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A356296 a(n) = Fibonacci(n)^2 mod n.

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

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