Ovidiu Bagdasar - cp's OEIS frontend

A345380 Number of Jacobsthal-Lucas numbers m <= n.

0, 1, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Ovidiu Bagdasar, Jun 16 2021

			a(0)=0 since the least term in A014551 is 1.
a(1)=1 since A014551(0) = 2 is followed in that sequence by 1.
a(k)=2 for 2 <= k <= 4 since the first terms of A014551 are {2, 1, 5}.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Tųrcąs, On some new results for the generalised Lucas sequences, An. Şt. Univ. Ovidius Constanţa (Romania, 2021) Vol. 29, No. 1, 17-36. See Section 5.6, pp. 35, Table 7.

Cf. A014551, A108852 (Fibonacci), A130245 (Lucas), A130253.

Block[{a = 1, b = -2, nn = 105, u, v = {}}, u = {2, a}; Do[AppendTo[u, Total[{-b, a} u[[-2 ;; -1]]]]; AppendTo[v, Count[u, _?(# <= i &)]], {i, nn}]; {Boole[First[u] <= 0]}~Join~v]  (* or *)
{0}~Join~Accumulate@ ReplacePart[ConstantArray[0, Last[#]], Map[# -> 1 &, #]] &@ LinearRecurrence[{1, 2}, {2, 1}, 8] (* Michael De Vlieger, Jun 16 2021 *)

A345379 Number of terms m <= n, where m is a term in the bisection of Lucas numbers (A005248).

Original entry on oeis.org

0, 0, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 0

Ovidiu Bagdasar, Jun 16 2021

			a(0)=a(1)=0, since the least term in A005248 is 2.
a(2)=1 since A005248(0) = 2 is followed in that sequence by 3.
a(k)=3 for 3 <= k <= 6 since the first terms of A005248 are {0, 2, 3, 7}.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Tųrcąs, On some new results for the generalised Lucas sequences, An. Şt. Univ. Ovidius Constanţa (Romania, 2021) Vol. 29, No. 1, 17-36. See Section 5.4, pp. 33-34, Table 4.

Cf. A005248, A108852 (Fibonacci), A130245 (Lucas), A130260.

Block[{a = 3, b = 1, nn = 105, u, v = {}}, u = {2, a}; Do[AppendTo[u, Total[{-b, a} u[[-2 ;; -1]]]]; AppendTo[v, Count[u, _?(# <= i &)]], {i, nn}]; {Boole[First[u] <= 0]}~Join~v] (* or *)
{0}~Join~Accumulate@ ReplacePart[ConstantArray[0, Last[#]], Map[# -> 1 &, #]] &@ LucasL@ Range[0, 10, 2] (* Michael De Vlieger, Jun 16 2021 *)

A345378 Number of terms m <= n, where m is a term in A006497.

Original entry on oeis.org

0, 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

Ovidiu Bagdasar, Jun 16 2021

Table 1 of Andrica 2021 paper (p. 24), refers to A006497 as "bronze Lucas" numbers.

			a(0)=a(1)=0, since the least term in A006497 is 2.
a(2)=1 since A006497(0) = 2 is followed in that sequence by 3.
a(k)=3 for 3 <= k <= 11 since the first terms of A006490 are {0, 2, 3, 11}.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Tųrcąs, On some new results for the generalised Lucas sequences, An. Şt. Univ. Ovidius Constanţa (Romania, 2021) Vol. 29, No. 1, 17-36. See Section 5.3, pp. 33, Table 4.

Cf. A006497, A108852 (Fibonacci), A130245 (Lucas), A345377.

Block[{a = 3, b = -1, nn = 105, u, v = {}}, u = {0, 1}; Do[AppendTo[u, Total[{-b, a} u[[-2 ;; -1]]]]; AppendTo[v, Count[u, _?(# <= i &)]], {i, nn}]; {Boole[First[u] <= 0]}~Join~v] (* or *)
{0}~Join~Accumulate@ ReplacePart[ConstantArray[0, Last[#]], Map[# -> 1 &, #]] &@ LucasL[Range[0, 4], 3] (* Michael De Vlieger, Jun 16 2021 *)

A345377 Number of terms m <= n, where m is a term in A006190.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 0

Ovidiu Bagdasar, Jun 16 2021

Table 1 of Andrica 2021 paper (p. 24) refers to A006190 as the "bronze Fibonacci" numbers.

			a(0)=1, since A006190(0) = 0 and A006190(1) = 1.
a(1)=a(2)=2 since 0 and 1 are the terms in A006190 that do not exceed 1 and 2, respectively.
a(k)=3 for 3 <= k <= 9 since the first terms of A006190 are {0, 1, 3, 10}.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Tųrcąs, On some new results for the generalised Lucas sequences, An. Şt. Univ. Ovidius Constanţa (Romania, 2021) Vol. 29, No. 1, 17-36. See Section 5.3, pp. 33, Table 4.

Cf. A006190, A108852 (Fibonacci), A130245 (Lucas), A345378.

Block[{a = 3, b = -1, nn = 105, u, v = {}}, u = {0, 1}; Do[AppendTo[u, Total[{-b, a} u[[-2 ;; -1]]]]; AppendTo[v, Count[u, _?(# <= i &)]], {i, nn}]; {Boole[First[u] <= 0]}~Join~v] (* or *)
Accumulate@ ReplacePart[ConstantArray[0, Last[#]], Map[# -> 1 &, # + 1]] &@ Fibonacci[Range[0, 5], 3] (* Michael De Vlieger, Jun 16 2021 *)

A345376 Number of Companion Pell numbers m <= n.

Original entry on oeis.org

0, 0, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

Ovidiu Bagdasar, Jun 16 2021

Table 1 of Andrica 2021 paper (p. 24) refers to A002203 as "Pell-Lucas" numbers.

			The Pell-Lucas numbers A002203 are 2, 2, 6, 14, 34, 82, ...
a(0)=a(1)=0, since there are no Pell-Lucas numbers less than or equal to 0 and 1, respectively.
a(2)=a(3)=a(4)=a(5)=2, since the first 2 Pell-Lucas numbers, 2 and 2, are less than or equal to 2, 3, 4, and 5, respectively.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Tųrcąs, On some new results for the generalised Lucas sequences, An. Şt. Univ. Ovidius Constanţa (Romania, 2021) Vol. 29, No. 1, 17-36. See Section 5.2, pp. 32-33, Table 3.

Cf. A002203, A108852 (Fibonacci), A130245 (Lucas), A335741 (Pell).

Block[{a = 2, b = -1, nn = 64, u, v = {}}, u = {2, a}; Do[AppendTo[u, Total[{-b, a} u[[-2 ;; -1]]]]; AppendTo[v, Count[u, ?(# <= i &)]], {i, nn}]; {Boole[First[u] <= 0]}~Join~v]  (* _Michael De Vlieger, Jun 16 2021 *)

A340236 Odd composite integers m such that A006190(3*m-J(m,13)) == 3 (mod m), where J(m,13) is the Jacobi symbol.

Original entry on oeis.org

9, 119, 121, 187, 327, 345, 649, 705, 1003, 1089, 1121, 1189, 1881, 2091, 2299, 3553, 4187, 5461, 5565, 5841, 6165, 6485, 7107, 7139, 7145, 7467, 7991, 8321, 8449, 11041, 12705, 12871, 13833, 14041, 16109, 16851

Offset: 1

Ovidiu Bagdasar, Jan 01 2021

nonn

The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy U(3*p-J(p,D)) == a (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4. The composite integers m with the property U(k*m-J(m,D)) == U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k- and parameter a.

Here b=-1, a=3, D=13 and k=3, while U(m) is A006190(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).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 2018, 24(1), 9--15.

Cf. A006190, A071904, A327653 (a=3, b=-1, k=1), A340119 (a=3, b=-1, k=2).

Cf. A340235 (a=1, b=-1, k=3), A340237 (a=5, b=-1, k=3), A340238 (a=7, b=-1, k=3).

Select[Range[3, 15000, 2], CoprimeQ[#, 13] && CompositeQ[#] && Divisible[Fibonacci[3*#-JacobiSymbol[#, 13], 3] - 3, #] &]

A340237 Odd composite integers m such that A052918(3*m-J(m,29)) == 5 (mod m), where J(m,29) is the Jacobi symbol.

Original entry on oeis.org

9, 27, 33, 35, 65, 81, 99, 121, 221, 243, 297, 363, 513, 585, 627, 705, 729, 891, 1089, 1539, 1541, 1881, 2145, 2187, 2299, 2673, 3267, 3605, 4181, 4573, 4579, 5265, 5633, 6721, 6993, 7865, 8019, 8979, 9131, 9801, 10307, 10877, 10881, 13333, 13741, 14001, 14705, 14989

Offset: 1

Ovidiu Bagdasar, Jan 01 2021

nonn

The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy U(3*p-J(p,D)) == a (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4.
The composite integers m with the property U(k*m-J(m,D)) == U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k- and parameter a.
Here b=-1, a=5, D=29 and k=3, while U(m) is A052918(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).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 2018, 24(1), 9--15.

Cf. A052918, A071904, A340095 (a=5, b=-1, k=1), A340120 (a=5, b=-1, k=2).

Cf. A340235 (a=1, b=-1, k=3), A340236 (a=3, b=-1, k=3), A340238 (a=7, b=-1, k=3).

Select[Range[3, 15000, 2], CoprimeQ[#, 29] && CompositeQ[#] && Divisible[Fibonacci[3*#-JacobiSymbol[#, 29], 5] - 5, #] &]

A340238 Odd composite integers m such that A054413(3*m-J(m,53)) == 7 (mod m), where J(m,53) is the Jacobi symbol.

Original entry on oeis.org

9, 25, 27, 51, 91, 105, 153, 185, 225, 289, 325, 425, 459, 481, 513, 747, 867, 897, 925, 945, 1001, 1189, 1299, 1469, 1633, 1785, 1921, 2241, 2245, 2599, 2601, 2651, 2769, 2907, 3051, 3277, 3825, 3897, 5681, 6225, 6507, 6777, 7225, 7361, 7803, 8023, 8227, 8701, 8721

Offset: 1

Ovidiu Bagdasar, Jan 01 2021

nonn

The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy U(3*p-J(p,D)) == a (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4.
The composite integers m with the property U(k*m-J(m,D)) == U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k- and parameter a.
Here b=-1, a=7, D=53 and k=3, while 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).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 2018, 24(1), 9--15.

Cf. A054413, A071904, A340096 (a=7, b=-1, k=1), A340121 (a=7, b=-1, k=2).

Cf. A340235 (a=1, b=-1, k=3), A340236 (a=3, b=-1, k=3), A340237 (a=5, b=-1, k=3).

Select[Range[3, 10000, 2], CoprimeQ[#, 53] && CompositeQ[#] && Divisible[Fibonacci[3*#-JacobiSymbol[#, 53], 7] - 7, #] &]

A340239 Odd composite integers m such that A001906(3m-J(m,5)) == 3J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

Original entry on oeis.org

9, 49, 63, 141, 161, 207, 323, 341, 377, 441, 671, 901, 1007, 1127, 1281, 1449, 1853, 1891, 2071, 2303, 2407, 2501, 2743, 2961, 3827, 4181, 4623, 5473, 5611, 5777, 6119, 6593, 6601, 6721, 7161, 7567, 8149, 8473, 8961, 9729, 9881

Offset: 1

Ovidiu Bagdasar, Jan 01 2021

nonn

The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy U(3*p-J(p,D)) == a*J(p,D) (mod p) whenever p is prime, k is a positive integer, b=1 and D=a^2-4.
The composite integers m with the property U(k*m-J(m,D)) == U(k-1)*J(m,D) (mod m) are called generalized Lucas pseudoprimes of level k+ and parameter a.
Here b=1, a=3, D=5 and k=3, while U(m) is A001906(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).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 2018, 24(1), 9--15.

Cf. A001906, A071904, A340097 (a=3, b=1, k=1), A340122 (a=3, b=1, k=2).

Cf. A340240 (a=5, b=1, k=3), A340241 (a=7, b=1, k=3).

Select[Range[3, 10000, 2], CoprimeQ[#, 5] && CompositeQ[#] &&  Divisible[ ChebyshevU[3*#  - JacobiSymbol[#, 5]  - 1, 3/2] - 3*JacobiSymbol[#, 5],  #] &]

A340240 Odd composite integers m such that A004254(3m-J(m,21)) == 5J(m,21) (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.

Original entry on oeis.org

55, 407, 527, 529, 551, 559, 965, 1199, 1265, 1633, 1807, 1919, 1961, 3401, 3959, 4033, 4381, 5461, 5777, 5977, 5983, 6049, 6233, 6439, 6479, 7141, 7195, 7645, 7999, 8639, 8695, 8993, 9265, 9361, 11663, 11989, 12209, 12265, 13019, 13021, 13199, 14023, 14465, 14491

Offset: 1

Ovidiu Bagdasar, Jan 01 2021

nonn

The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy U(3*p-J(p,D)) == a*J(p,D) (mod p) whenever p is prime, k is a positive integer, b=1 and D=a^2-4.
The composite integers m with the property U(k*m-J(m,D)) == U(k-1)*J(m,D) (mod m) are called generalized Lucas pseudoprimes of level k+ and parameter a.
Here b=1, a=5, D=21 and k=3, while U(m) is A004254(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).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 2018, 24(1), 9--15.

Cf. A004254, A071904, A340098 (a=5, b=1, k=1), A340123 (a=5, b=1, k=2).

Cf. A340239 (a=3, b=1, k=3), A340241 (a=7, b=1, k=3).

Select[Range[3, 15000, 2], CoprimeQ[#, 21] && CompositeQ[#] &&  Divisible[ ChebyshevU[3*# - JacobiSymbol[#, 21] - 1, 5/2] - 5*JacobiSymbol[#, 21],  #] &]

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User: Ovidiu Bagdasar

A345380 Number of Jacobsthal-Lucas numbers m <= n.

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A345379 Number of terms m <= n, where m is a term in the bisection of Lucas numbers (A005248).

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A345378 Number of terms m <= n, where m is a term in A006497.

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A345377 Number of terms m <= n, where m is a term in A006190.

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A345376 Number of Companion Pell numbers m <= n.

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A340236 Odd composite integers m such that A006190(3*m-J(m,13)) == 3 (mod m), where J(m,13) is the Jacobi symbol.

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A340237 Odd composite integers m such that A052918(3*m-J(m,29)) == 5 (mod m), where J(m,29) is the Jacobi symbol.

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A340238 Odd composite integers m such that A054413(3*m-J(m,53)) == 7 (mod m), where J(m,53) is the Jacobi symbol.

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A340239 Odd composite integers m such that A001906(3m-J(m,5)) == 3J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

A340240 Odd composite integers m such that A004254(3m-J(m,21)) == 5J(m,21) (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.