A337235 -id:A337235 - cp's OEIS frontend

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

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, #] &]

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, #] &]

A338313 Even composite positive integers m such that A052918(m-1)^2 == 1 (mod m).

Original entry on oeis.org

4, 8, 16, 32, 68, 248, 268, 544, 1328, 4216, 4768, 9112, 9376, 12664, 20128, 22112, 24536, 25544, 30488, 43262, 61574, 125792, 148004, 304792, 398248, 493646, 648848, 913456, 1036664, 1975784, 2350792, 3672454, 4248488, 5422688, 6318188, 6768928, 7079656, 8560724

Offset: 1

Ovidiu Bagdasar, Oct 22 2020

nonn

If p is a prime, then A052918(p-1)^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(m+2) = a*U(m+1)-b*U(m) 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=5, b=-1, U(m) recovers A052918(m-1), for m=1,2,....

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. A337235 (a=3)

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

More terms from Amiram Eldar, Oct 22 2020

a(31)-a(38) from Daniel Suteu, Oct 22 2020

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

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A338313 Even composite positive integers m such that A052918(m-1)^2 == 1 (mod m).

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