A340124 Odd composite integers m such that A004187(2*m-J(m,45)) == J(m,45) (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol.

49, 323, 329, 343, 377, 451, 1081, 1127, 1771, 1819, 1891, 2033, 2303, 2401, 3653, 3827, 4181, 5671, 5777, 6601, 6721, 7471, 7931, 8149, 8557, 9691, 10877, 11309, 11663, 13201, 13861, 13981, 14701, 15251, 15449, 16121, 16807, 17119, 17513, 17687, 17711

Offset

1,1

Comments

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(2*p-J(p,D)) == 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)) == 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=45 and k=2, while U(m) is A004187(m).

References

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).

Links

Table of n, a(n) for n=1..41.

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.

Mathematica

Select[Range[3, 20000, 2], CoprimeQ[#, 45] && CompositeQ[#] &&
Divisible[ ChebyshevU[2*# - JacobiSymbol[#, 45] - 1, 7/2] - JacobiSymbol[#, 45],  #] &]

Crossrefs

Cf. A004187, A071904, A340099 (a=7, b=1, k=1).

Cf. A340122 (a=3, b=1, k=2), A340123 (a=5, b=1, k=2).

Sequence in context: A251222 A250967 A245033 * A017474 A335389 A036318

Adjacent sequences: A340121 A340122 A340123 * A340125 A340126 A340127

Keyword

nonn

Author

Ovidiu Bagdasar, Dec 28 2020

Status

approved