A339725 - OEIS

A339725

Odd composite integers m such that A006497(3*m-J(m,13)) == 11*J(m,13) (mod m), where J(m,13) is the Jacobi symbol.

9, 27, 119, 133, 145, 165, 205, 261, 341, 393, 649, 693, 705, 901, 945, 1121, 1173, 1189, 1353, 1431, 1485, 1881, 2133, 2805, 3201, 3605, 3745, 4187, 5173, 5461, 5841, 5945, 6165, 6213, 6485, 6943, 6993, 7107, 7991, 8321, 8449, 9669, 11041, 11781, 11961, 12861

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OFFSET

1,1

COMMENTS

The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy V(k*p-J(p,D)) == V(k-1)*J(p,D) (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 V(k*m-J(m,D)) == V(k-1)*J(m,D) (mod m) are called generalized Pell-Lucas pseudoprimes of level k- and parameter a.

Here b=-1, a=3, D=13 and k=3, while V(m) recovers A006497(m), with V(2)=11.

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

Table of n, a(n) for n=1..46.

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 24(1), 9-15 (2018).

MATHEMATICA

Select[Range[3, 13000, 2], CoprimeQ[#, 13] && CompositeQ[#] && Divisible[LucasL[3*# - JacobiSymbol[#, 13], 3] - 11*JacobiSymbol[#, 13], #] &]

CROSSREFS

Cf. A006497, A071904, A339126 (a=3, b=-1, k=1), A339518 (a=3, b=-1, k=2).

Cf. A339724 (a=1, b=-1), A339726 (a=5, b=-1), A339727 (a=7, b=-1).

Sequence in context: A053762 A126322 A020279 * A328604 A357667 A230185

Adjacent sequences: A339722 A339723 A339724 * A339726 A339727 A339728

KEYWORD

nonn

AUTHOR

Ovidiu Bagdasar, Dec 14 2020

STATUS

approved