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A339128 Odd composite integers m such that A086902(m-J(m,53)) == 2*J(m,53) (mod m), where J(m,53) is the Jacobi symbol. 8
9, 25, 49, 51, 91, 121, 125, 153, 169, 289, 325, 361, 441, 529, 625, 637, 833, 841, 867, 961, 1183, 1225, 1369, 1633, 1681, 1849, 1921, 2209, 2599, 2601, 2651, 3481, 3721, 4225, 4489, 4625, 5041, 5125, 5329, 5537, 6241, 6889, 7225, 7267, 7497, 7921, 8125, 8281 (list; graph; refs; listen; history; text; internal format)
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 the identity

V(p-J(p,D)) == 2*J(p,D) (mod p) when p is prime, b=-1 and D=a^2+4.

This sequence has the odd composite integers with V(m-J(m,D)) == 2*J(m,D) (mod m).

For a=7 and b=-1, we have D=53 and V(m) recovers A086902(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..48.

MATHEMATICA

Select[Range[3, 10000, 2], CompositeQ[#] && Divisible[LucasL[# - (j = JacobiSymbol[#, 53]), 7] - 2*j, #] &] (* Amiram Eldar, Nov 26 2020 *)

CROSSREFS

Cf. A086902.

Cf. A339125 (a=1, b=-1), A339126 (a=3, b=-1), A339127 (a=5, b=-1), A339129 (a=3, b=1), A339130 (a=5, b=1), A339131 (a=7, b=1).

Sequence in context: A247687 A075026 A339727 * A113659 A325701 A113745

Adjacent sequences:  A339125 A339126 A339127 * A339129 A339130 A339131

KEYWORD

nonn

AUTHOR

Ovidiu Bagdasar, Nov 24 2020

EXTENSIONS

More terms from Amiram Eldar, Nov 26 2020

STATUS

approved

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Last modified April 18 12:45 EDT 2021. Contains 343088 sequences. (Running on oeis4.)