A340096 Odd composite integers m such that A054413(m-J(m,53)) == 0 (mod m), where J(m,53) is the Jacobi symbol.

25, 35, 51, 65, 91, 175, 325, 391, 455, 575, 1247, 1295, 1633, 1763, 1775, 1921, 2275, 2407, 2599, 2651, 3367, 4199, 4579, 4623, 5629, 6441, 9959, 10465, 10825, 10877, 12025, 13021, 15155, 16021, 18881, 19019, 19039, 19307, 19669

Offset: 1

Author

Ovidiu Bagdasar, Dec 28 2020

Keywords

• nonn

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 the identity

U(p-J(p,D)) == 0 (mod p) when p is prime, b=-1 and D=a^2+4.

This sequence contains the odd composite integers with U(m-J(m,D)) == 0 (mod m).

For a=7 and b=-1, we have D=53 and U(m) recovers A054413(m).

If even numbers greater than 2 that are coprime to 53 are allowed, then 10, 50, 370, 5050, ... would also be terms. - Jianing Song, Jan 09 2021

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.

Crossrefs

Cf. A054413, A071904, A081264 (a=1, b=-1), A327653 (a=3,b=-1), A340095 (a=5, b=-1)

Cf. A340097 (a=3, b=1), A340098 (a=5, b=1), A340099 (a=7, b=1).

Programs

Mathematica

Select[Range[3,20000, 2], CoprimeQ[#, 53] && CompositeQ[#] && Divisible[Fibonacci[#-JacobiSymbol[#, 53], 7], #] &]