A337625 Odd composite integers m such that F(m)^2 == 1 (mod m) and L(m) == 1 (mod m), where F(m) and L(m) are the m-th Fibonacci and Lucas numbers, respectively.

2737, 4181, 5777, 6721, 10877, 13201, 15251, 29281, 34561, 51841, 64079, 64681, 67861, 68251, 75077, 80189, 90061, 96049, 97921, 100127, 105281, 113573, 118441, 146611, 161027, 162133, 163081, 179697, 186961, 194833, 197209, 219781, 228241, 231703, 252601, 254321

Offset: 1

Author

Ovidiu Bagdasar, Sep 19 2020

Keywords

• nonn

Comments

Intersection of A005845 and A337231.

These numbers may be called weak generalized Fibonacci-Lucas-Bruckner pseudoprimes.

If p is a prime, then F(p)^2 == 1 (mod p) and L(p) == 1 (mod p).

This sequence contains the odd composite integers for which these congruences hold.

For a,b integers, the following sequences are defined:

generalized Lucas sequences by U(n+2)=a*U(n+1)-b*U(n) and U(0)=0, U(1)=1,

generalized Pell-Lucas sequences by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a.

These satisfy the identities U(p)^2 == 1 and V(p)==a (mod p) for p prime and b=1,-1.

These numbers may be called weak generalized Lucas-Bruckner pseudoprimes of parameters a and b.The current sequence is defined for a=1 and b=-1.

Examples: a(n) is also the number of Jones graphs on n nodes.

Links

• Dorin Andrica and Ovidiu Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math., 18 (2021), 47.

Crossrefs

Cf. A005845 and A337231.

Programs

Mathematica

Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 1]*Fibonacci[#, 1] - 1, #] && Divisible[LucasL[#, 1] - 1, #] &]

Extensions

More terms from Amiram Eldar, Sep 19 2020