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

1,1

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)

D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021)

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.

Mathematica

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

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).

Sequence in context: A164384 A138193 A330947 * A036315 A340120 A020154

Adjacent sequences: A337234 A337235 A337236 * A337238 A337239 A337240

Keyword

nonn

Author

Ovidiu Bagdasar, Aug 20 2020

Status

approved