A337234 - OEIS

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A337234

Odd composite integers m such that A006190(m)^2 == 1 (mod m).

9, 33, 55, 63, 99, 119, 153, 231, 385, 399, 561, 649, 935, 981, 1023, 1071, 1179, 1189, 1199, 1441, 1595, 1763, 1881, 1953, 2001, 2065, 2255, 2289, 2465, 2703, 2751, 2849, 2871, 3519, 3599, 3655, 3927, 4059, 4081, 4187, 5015, 5151, 5559, 6061, 6119, 6215, 6273, 6431

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OFFSET

1,1

COMMENTS

If p is a prime, then A006190(p)^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.

For a=3, b=-1, U(n) recovers A006190(n) ("Bronze" Fibonacci numbers).

REFERENCES

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

LINKS

Table of n, a(n) for n=1..48.

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).

MATHEMATICA

Select[Range[3, 25000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 3]*Fibonacci[#, 3] - 1, #] &]

CROSSREFS

Cf. A337231 (a=1, odd terms), A337232 (a=1, even terms), A337233 (a=2).

Sequence in context: A280405 A111351 A065064 * A075812 A256868 A146262

Adjacent sequences: A337231 A337232 A337233 * A337235 A337236 A337237

KEYWORD

nonn

AUTHOR

Ovidiu Bagdasar, Aug 20 2020

STATUS

approved