A340099 Odd composite integers m such that A004187(m-J(m,45)) == 0 (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol.

323, 329, 377, 451, 1081, 1771, 1819, 1891, 2033, 3653, 3827, 4181, 5671, 5777, 6601, 6721, 7471, 7931, 8149, 8557, 10877, 11309, 11663, 13201, 13861, 13981, 14701, 15251, 15449, 17119, 17513, 17687, 17711, 17941, 18407, 19043, 19951, 20447, 20473, 23407, 23771, 23851, 23999

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=45 and U(m) recovers 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

• 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. A004187, A071904, A081264 (a=1, b=-1), A327653 (a=3,b=-1), A340095 (a=5, b=-1), A340096 (a=7, b=-1), A340097 (a=3, b=1), A340098 (a=5, b=1).

Programs

Mathematica

Select[Range[3, 25000, 2], CoprimeQ[#, 45] && CompositeQ[#] && Divisible[ChebyshevU[# - JacobiSymbol[#, 45] - 1, 7/2], #] &]