A339521 -id:A339521 - cp's OEIS frontend

A339522 Odd composite integers m such that A003501(2*m-J(m,21)) == 5 (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.

95, 115, 145, 253, 391, 527, 551, 713, 715, 779, 935, 1045, 1615, 1705, 1805, 1807, 1919, 2185, 2627, 2755, 2893, 2929, 2945, 3281, 4033, 4141, 4205, 5191, 5671, 5777, 5983, 6049, 6479, 7645, 7739, 8441, 8555, 8695, 9361, 11663, 11815, 12121, 12209, 12265, 14491

Offset: 1

Ovidiu Bagdasar, Dec 07 2020

nonn

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 V(k*p-J(p,D)) == V(k-1) (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 V(k*m-J(m,D)) == V(k-1) (mod m) are called generalized Pell-Lucas pseudoprimes of level k+ and parameter a.
Here b=1, a=5, D=21 and k=2, while V(m) recovers A003501(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).

Robert Israel, Table of n, a(n) for n = 1..1000
Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 24(1), 9-15 (2018).

Cf. A003501, A071904, A339130 (a=5, b=1, k=1).

Cf. A339521 (a=3, b=1), A339523 (a=7, b=1).

filter:= proc(m)
uses LinearAlgebra:-Modular;
local p,M;
  if igcd(m,21) <> 1 then return false fi;
  if isprime(m) then return false fi;
  p:= 2*m - numtheory:-jacobi(m,21);
  M:= Mod(m,[[0,1],[-1,5]],integer[8]);
  (MatrixPower(m,M,p) . <2,5>)[1] - 5 mod m = 0
end proc:
select(filter, [seq(i,i=9..20000,2)]); # Robert Israel, Dec 15 2020

Select[Range[3, 20000, 2], CoprimeQ[#, 21] && CompositeQ[#] && Divisible[2*ChebyshevT[2*# - JacobiSymbol[#, 21], 5/2] - 5, #] &]

A339523 Odd composite integers m such that A056854(2*m-J(m,45)) == 7 (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol.

Original entry on oeis.org

91, 203, 323, 329, 377, 451, 1001, 1081, 1183, 1547, 1729, 1771, 1819, 1891, 1967, 2033, 2093, 2639, 2821, 3197, 3311, 3653, 3731, 3827, 4181, 4669, 5551, 5671, 5777, 5887, 6601, 6721, 7471, 7931, 7973, 8149, 8557, 9541, 9737, 10877, 11309, 11663, 11977, 13201

Offset: 1

Ovidiu Bagdasar, Dec 07 2020

nonn

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 V(k*p-J(p,D)) == V(k-1) (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 V(k*m-J(m,D)) == V(k-1) (mod m) are called generalized Pell-Lucas pseudoprimes of level k+ and parameter a.
Here b=1, a=7, D=45 and k=2, while V(m) recovers A056854(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).

Robert Israel, Table of n, a(n) for n = 1..1000
Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 24(1), 9-15 (2018).

Cf. A056854, A071904, A339131 (a=7, b=1, k=1).

Cf. A339521 (a=3, b=1), A339522 (a=5, b=1).

filter:= proc(m)
uses LinearAlgebra:-Modular;
local p,M;
  if igcd(m,45) <> 1 then return false fi;
  if isprime(m) then return false fi;
  p:= 2*m - numtheory:-jacobi(m,45);
  M:= Mod(m,[[0,1],[-1,7]],integer[8]);
  (MatrixPower(m,M,p) . <2,7>)[1] - 7 mod m = 0
end proc:
select(filter, [seq(i,i=9..10000,2)]); # Robert Israel, Dec 15 2020

Select[Range[3, 15000, 2], CoprimeQ[#, 45] && CompositeQ[#] && Divisible[LucasL[4*(2*# - JacobiSymbol[#, 45])] - 7, #] &]

A339728 Odd composite integers m such that A005248(3*m-J(m,5)) == 7 (mod m), where J(m,5) is the Jacobi symbol.

Original entry on oeis.org

9, 21, 27, 63, 161, 189, 207, 261, 287, 323, 341, 377, 671, 783, 861, 901, 987, 1007, 1107, 1269, 1281, 1287, 1449, 1853, 1891, 2071, 2241, 2407, 2431, 2501, 2529, 2567, 2743, 2961, 3201, 3827, 4181, 4623, 5029, 5473, 5611, 5777, 5781, 6119, 6601, 6721, 7161

Offset: 1

Ovidiu Bagdasar, Dec 14 2020

nonn

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 V(k*p-J(p,D)) == V(k-1) (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 V(k*m-J(m,D)) == V(k-1) (mod m) are called generalized Pell-Lucas pseudoprimes of level k+ and parameter a.
Here b=1, a=3, D=5 and k=3, while V(m) recovers A005248(m), with V(2)=7.

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).

Amiram Eldar, Table of n, a(n) for n = 1..1000
Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 24(1), 9-15 (2018).

Cf. A005248, A071904, A339129 (a=3, b=1, k=1), A339521 (a=3, b=1, k=2).

Cf. A339729 (a=5, b=1), A339730 (a=7, b=1).

Select[Range[3, 7500, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[LucasL[2*(3*# - JacobiSymbol[#, 5])] - 7, #] &]

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A339522 Odd composite integers m such that A003501(2*m-J(m,21)) == 5 (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.

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A339523 Odd composite integers m such that A056854(2*m-J(m,45)) == 7 (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol.

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Crossrefs

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Maple

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A339728 Odd composite integers m such that A005248(3*m-J(m,5)) == 7 (mod m), where J(m,5) is the Jacobi symbol.

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Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica