A135535 Primes of the form 4^k - 3.

13, 61, 1021, 4093, 16381, 1048573, 4194301, 16777213, 19807040628566084398385987581, 83076749736557242056487941267521533, 5316911983139663491615228241121378301, 1427247692705959881058285969449495136382746621, 23945242826029513411849172299223580994042798784118781, 118571099379011784113736688648896417641748464297615937576404566024103044751294461, 139984046386112763159840142535527767382602843577165595931249318810236991948760059086304843329475444733

Offset: 1

Author

Daniele Corradetti (d.corradetti(AT)gmail.com), Feb 21 2008

Keywords

• nonn

Comments

Involved in the "New Mersenne Prime Conjecture" and in some generalizations of Mersenne primes.

Subsequence of A050415. - Elmo R. Oliveira, Nov 28 2023

Examples

			16381 is a term because 4^7 - 3 = 16381 is prime.
		

References

• Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (pp. 114-134).

Links

• P. T. Bateman, J. L. Selfridge and S. S. Wagstaff, Jr., The New Mersenne Conjecture, Amer. Math. Monthly 96, 125-128, 1989.
• D. Minoli and Robert Bear, Hyperperfect Numbers, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157. [Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009]
• Daniel Minoli and W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing. [Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009]
• Paul Tannery, Questions 659 et 660, L'Intermédiaire des mathématiciens, Tome II (1895) p. 317.
• Eric Weisstein's World of Mathematics, New Mersenne Prime Conjecture

Crossrefs

Cf. A000040, A050415, A057732, A057733, A059266.

Programs

Mathematica

Do[If[PrimeQ[4^n - 3], Print[4^n - 3]], {n, 100}] (* Robert G. Wilson v, Feb 29 2008 *)
Select[4^Range[200]-3,PrimeQ] (* Harvey P. Dale, Jul 11 2022 *)

Formula

a(n) = 4^A059266(n) - 3. - Ryan Propper, Feb 26 2008

Extensions

More terms from R. J. Mathar, Robert G. Wilson v and Ryan Propper, Feb 26 2008