A337626 -id:A337626 - cp's OEIS frontend

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.

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

Ovidiu Bagdasar, Sep 19 2020

nonn

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.

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

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

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

More terms from Amiram Eldar, Sep 19 2020

A337628 Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 5 (mod m), where U(m) and V(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=5 and b=-1, respectively.

Original entry on oeis.org

9, 27, 65, 121, 385, 533, 1035, 4081, 5089, 5993, 6721, 7107, 10877, 11285, 13281, 13741, 14705, 16721, 18901, 19601, 19951, 20705, 24769, 25345, 26599, 26937, 28741, 29161, 32639, 37949, 39185, 39985, 45305, 45451, 49105, 50553, 51085, 52801, 57205, 64297, 72385

Offset: 1

Ovidiu Bagdasar, Sep 19 2020

nonn

Intersection of A335671 and A337237.
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=5 and b=-1.

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

Cf. A335671 and A337237.

Similar sequences: A337625 (a=1), A337626 (a=3) and A337627 (a=4).

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

More terms from Amiram Eldar, Sep 19 2020

A337777 Even composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 3 (mod m), where U(m)=A001906(m) and V(m)=A005248(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=3 and b=1, respectively.

Original entry on oeis.org

4, 44, 836, 1364, 2204, 7676, 7964, 9164, 11476, 12524, 23804, 31124, 32642, 39556, 73124, 80476, 99644, 110564, 128876, 156484, 192676, 199924, 287804, 295196, 315524, 398924, 542242, 715604, 780044, 934876, 987524, 1050524, 1339516, 1390724, 1891124, 1996796

Offset: 1

Ovidiu Bagdasar, Sep 20 2020

nonn

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=3 and b=1.

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

Cf. A337626.

Select[Range[2, 20000, 2], CompositeQ[#] && Divisible[LucasL[2#] - 3, #] && Divisible[ChebyshevU[#-1, 3/2]*ChebyshevU[#-1, 3/2] - 1, #] &]

More terms from Amiram Eldar, Sep 21 2020

A337629 Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 6 (mod m), where U(m) and V(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=6 and b=-1, respectively.

Original entry on oeis.org

57, 481, 629, 721, 779, 1121, 1441, 1729, 2419, 2737, 6721, 7471, 8401, 9361, 10561, 11521, 11859, 12257, 15281, 16321, 16583, 18849, 24721, 25441, 25593, 33649, 35219, 36481, 36581, 37949, 39169, 41041, 45961, 46999, 50681, 52417, 53041, 53521, 54757, 55537

Offset: 1

Ovidiu Bagdasar, Sep 19 2020

nonn

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=6 and b=-1.

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

Cf. A337625 (a=1), A337626 (a=3), A337627 (a=4), A337628 (a=5).

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

More terms from Amiram Eldar, Sep 19 2020

A337630 Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 7 (mod m), where U(m) and V(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=7 and b=-1, respectively.

Original entry on oeis.org

25, 51, 91, 161, 325, 425, 561, 791, 1105, 1633, 1921, 2001, 2465, 2599, 2651, 2737, 7345, 8449, 9361, 10325, 10465, 10825, 11285, 12025, 12291, 13021, 15457, 17111, 18193, 18881, 19307, 20705, 20833, 21931, 24081, 24661, 31521, 32305, 37925, 38801, 39059, 40641

Offset: 1

Ovidiu Bagdasar, Sep 19 2020

nonn

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=7 and b=-1.

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

Cf. A337625 (a=1), A337626 (a=3), A337627 (a=4), A337628 (a=5), A337629 (a=6).

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

More terms from Amiram Eldar, Sep 19 2020

cp's OEIS Frontend

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.

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A337628 Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 5 (mod m), where U(m) and V(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=5 and b=-1, respectively.

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A337777 Even composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 3 (mod m), where U(m)=A001906(m) and V(m)=A005248(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=3 and b=1, respectively.

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Mathematica

Extensions

A337629 Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 6 (mod m), where U(m) and V(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=6 and b=-1, respectively.

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Programs

Mathematica

Extensions

A337630 Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 7 (mod m), where U(m) and V(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=7 and b=-1, respectively.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions