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

1,1

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 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

Table of n, a(n) for n=1..43.

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.

Mathematica

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

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).

Sequence in context: A001233 A145148 A214099 * A237856 A016260 A011810

Adjacent sequences: A340094 A340095 A340096 * A340098 A340099 A340100

Keyword

nonn

Author

Ovidiu Bagdasar, Dec 28 2020

Extensions

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

Status

approved