A340238 - OEIS

A340238

Odd composite integers m such that A054413(3*m-J(m,53)) == 7 (mod m), where J(m,53) is the Jacobi symbol.

9, 25, 27, 51, 91, 105, 153, 185, 225, 289, 325, 425, 459, 481, 513, 747, 867, 897, 925, 945, 1001, 1189, 1299, 1469, 1633, 1785, 1921, 2241, 2245, 2599, 2601, 2651, 2769, 2907, 3051, 3277, 3825, 3897, 5681, 6225, 6507, 6777, 7225, 7361, 7803, 8023, 8227, 8701, 8721

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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 U(3*p-J(p,D)) == a (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 U(k*m-J(m,D)) == U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k- and parameter a.

Here b=-1, a=7, D=53 and k=3, while U(m) is A054413(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

Table of n, a(n) for n=1..49.

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, 10000, 2], CoprimeQ[#, 53] && CompositeQ[#] && Divisible[Fibonacci[3*#-JacobiSymbol[#, 53], 7] - 7, #] &]

CROSSREFS

Cf. A054413, A071904, A340096 (a=7, b=-1, k=1), A340121 (a=7, b=-1, k=2).

Cf. A340235 (a=1, b=-1, k=3), A340236 (a=3, b=-1, k=3), A340237 (a=5, b=-1, k=3).

Sequence in context: A339127 A117580 A280609 * A020308 A108989 A352492

Adjacent sequences: A340235 A340236 A340237 * A340239 A340240 A340241

KEYWORD

nonn

AUTHOR

Ovidiu Bagdasar, Jan 01 2021

STATUS

approved