login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A354168 Let M_p = 2^p-1 be a Mersenne prime, where p is an odd prime. Sequence lists p such that b_{p-2} == -2^((p+1)/2) mod M_p, where {b_k} is defined in the Comments. 1
7, 17, 19, 89, 107, 521, 607, 1279, 2281, 3217, 4423, 9689, 11213, 21701, 44497, 216091, 859433, 1257787, 24036583, 30402457, 32582657, 42643801, 57885161, 74207281, 82589933 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Let M_p = 2^p-1 (not necessarily a prime) where p is an odd prime, and define b_1 = 4; b_k = b_{k-1}^2 - 2 (mod M_p) for k >= 2.

The Lucas-Lehmer theorem says that M_p is a prime iff b_{p-1} == 0 (mod M_p).

Furthermore, if M_p is a prime, then b_{p-2} is congruent to +- 2^((p+1)/2) (mod M_p).

This partitions the Mersenne prime exponents A000043 into two classes, listed here and in A354167.

REFERENCES

J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 16.1.

LINKS

Table of n, a(n) for n=1..25.

Mersenneforum, data for all known Mersenne penultimate residues (up to M#51)

Wikipedia, Lucas-Lehmer primality test. Sign of penultimate term

CROSSREFS

Cf. A000043, A000668, A354167.

Cf. A123271 (sign of the penultimate term of the Lucas-Lehmer sequence).

Sequence in context: A155774 A180641 A234095 * A287182 A104480 A053746

Adjacent sequences:  A354165 A354166 A354167 * A354169 A354170 A354171

KEYWORD

nonn,more

AUTHOR

N. J. A. Sloane, Jun 02 2022, based on Section 16.1 of Cosgrave (2022).

EXTENSIONS

Thanks to Chai Wah Wu for several corrections. - N. J. A. Sloane, Jun 02 2022

a(16) from Chai Wah Wu, Jun 03 2022

a(17)-a(18) from Chai Wah Wu, Jun 04 2022

a(19)-a(25) from Serge Batalov, _Jun 11 2022

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 4 16:09 EDT 2022. Contains 357239 sequences. (Running on oeis4.)