A338081 Odd composite integers such that A054413(m)^2 == 1 (mod m).

21, 25, 35, 49, 51, 65, 85, 91, 119, 147, 161, 175, 221, 231, 245, 325, 357, 377, 391, 399, 425, 455, 539, 559, 561, 575, 595, 629, 637, 759, 791, 833, 1001, 1105, 1127, 1225, 1247, 1295, 1309, 1495, 1547, 1633, 1763, 1775, 1921, 2001, 2015, 2261, 2275, 2407

Offset: 1

Author

Ovidiu Bagdasar, Oct 08 2020

Keywords

• nonn

Comments

The generalized Lucas sequence of integer parameters (a,b) is defined by

U(m+2) = a*U(m+1)-b*U(m) and U(0)=0, U(1)=1.

Whenever p is prime and b=-1,1 we have U^2(p) == 1 (mod p).

Here we define the odd composite integers for which U^2(m) == 1 (mod m) holds, for a=7, b=-1, where U(m) is A054413(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)

Links

• Dorin Andrica and Ovidiu Bagdasar, On Generalized Lucas Pseudoprimality of Level k, Mathematics (2021) Vol. 9, 838.

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), A337237 (a=5), A338081 (a=6).

Programs

Mathematica

Select[Range[3, 15000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 7]*Fibonacci[#, 7] - 1, #] &]