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A340121
Odd composite integers m such that A054413(2*m-J(m,53)) == 1 (mod m), where J(m,53) is the Jacobi symbol.
4
25, 35, 39, 49, 51, 65, 91, 147, 175, 245, 301, 325, 343, 391, 455, 507, 575, 605, 637, 663, 741, 833, 897, 903, 935, 1127, 1205, 1225, 1247, 1295, 1505, 1595, 1633, 1715, 1763, 1775, 1911, 1921, 2107, 2275, 2401, 2407, 2499, 2599, 2651, 3025, 3143, 3185, 3311
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)) == 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 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=2, while U(m) is A054413(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
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, 10000, 2], CoprimeQ[#, 53] && CompositeQ[#] && Divisible[Fibonacci[2*#-JacobiSymbol[#, 53], 7] - 1, #] &]
CROSSREFS
Cf. A054413, A071904, A340096 (a=7, b=-1, k=1).
Cf. A340118 (a=1, b=-1, k=2), A340119 (a=3, b=-1, k=2), A340120 (a=5, b=-1, k=2).
Sequence in context: A098368 A281370 A101066 * A061442 A049514 A049518
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Dec 28 2020
STATUS
approved