A340097 Odd composite integers m such that A001906(m-J(m,5)) == 0 (mod m) and gcd(m,5)=1, where J(m,5) is the Jacobi symbol.

21, 323, 329, 377, 451, 861, 1081, 1819, 1891, 2033, 2211, 3653, 3827, 4089, 4181, 5671, 5777, 6601, 6721, 8149, 8557, 10877, 11309, 11663, 13201, 13861, 13981, 14701, 15251, 17119, 17513, 17711, 17941, 18407, 19043, 19951, 20473, 23407, 25369, 25651, 25877, 27323, 27511

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=3 and b=1, we have D=5 and U(m) recovers A001906(m).

References

• D. Andrica and O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.

Links

• D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math., 18, 47 (2021).
• D. Andrica and O. Bagdasar, On generalized pseudoprimality of level k, Mathematics 2021, 9(8), 838.
• 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. A001906, A071904, A081264 (a=1, b=-1), A327653 (a=3,b=-1), A340095 (a=5, b=-1), A340096 (a=7, b=-1), A340098 (a=5, b=1), A340099 (a=7, b=1).

Programs

Mathematica

Select[Range[3, 30000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[ChebyshevU[# - JacobiSymbol[#, 5] - 1, 3/2], #] &]

Extensions

Coprime condition added to definition by Georg Fischer, Jul 20 2022