A337237 Odd composite integers such that A052918(m-1)^2 == 1 (mod m).

9, 15, 25, 27, 35, 45, 65, 75, 91, 121, 135, 143, 175, 225, 275, 325, 385, 455, 533, 595, 615, 675, 935, 1035, 1107, 1325, 1359, 1431, 1495, 1547, 1573, 1935, 2015, 2255, 2275, 2775, 3025, 3059, 3575, 3605, 4025, 4081, 4235, 4355, 5005, 5089, 5475, 5525, 5719, 5993, 6165

Offset: 1

Author

Ovidiu Bagdasar, Aug 20 2020

Keywords

• nonn

Comments

If p is a prime, then A052918(p-1)^2 == 1 (mod p).

This sequence contains the odd composite integers for which the congruence holds.

The generalized Lucas sequence of integer parameters (a,b) defined by U(n+2) = a*U(n+1)-b*U(n) and U(0)=0, U(1)=1, satisfies the identity U^2(p) == 1 (mod p) whenever p is prime and b=-1,1 (this property is a form of pseudoprimality).

For a=5, b=-1, U(n) recovers A052918(n-1), for n=1,2,....

References

• D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000
• Dorin Andrica and Ovidiu Bagdasar, On Generalized Lucas Pseudoprimality of Level k, Mathematics (2021) Vol. 9, 838.
• D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, preprint for Mediterr. J. Math. 18, 47 (2021).

Crossrefs

Cf. A337231 (a=1, odd terms), A337232 (a=1, even terms), A337233 (a=2), A337234 (a=3, odd terms), A337235 (a=3, even terms), A337236 (a=4).

Programs

Mathematica

Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 5]*Fibonacci[#, 5] - 1, #] &]