cp's OEIS Frontend

A135535 Primes of the form 4^k - 3.

%I A135535 #27 Feb 16 2025 08:33:07
%S A135535 13,61,1021,4093,16381,1048573,4194301,16777213,
%T A135535 19807040628566084398385987581,83076749736557242056487941267521533,
%U A135535 5316911983139663491615228241121378301,1427247692705959881058285969449495136382746621,23945242826029513411849172299223580994042798784118781,118571099379011784113736688648896417641748464297615937576404566024103044751294461,139984046386112763159840142535527767382602843577165595931249318810236991948760059086304843329475444733
%N A135535 Primes of the form 4^k - 3.
%C A135535 Involved in the "New Mersenne Prime Conjecture" and in some generalizations of Mersenne primes.
%C A135535 Subsequence of A050415. - _Elmo R. Oliveira_, Nov 28 2023
%D A135535 Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (pp. 114-134).
%H A135535 P. T. Bateman, J. L. Selfridge and S. S. Wagstaff, Jr., <a href="http://www.jstor.org/stable/2323195">The New Mersenne Conjecture</a>, Amer. Math. Monthly 96, 125-128, 1989.
%H A135535 D. Minoli and Robert Bear, <a href="http://www.pme-math.org/journal/issues/PMEJ.Vol.6.No.3.pdf">Hyperperfect Numbers</a>, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157. [Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009]
%H A135535 Daniel Minoli and W. Nakamine, <a href="http://dx.doi.org/10.1109/ICASSP.1980.1170906">Mersenne Numbers Rooted On 3 For Number Theoretic Transforms</a>, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing. [Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009]
%H A135535 Paul Tannery, <a href="https://books.google.fr/books?id=F8Q_AQAAIAAJ&amp;printsec=frontcover&amp;hl=fr#v=onepage&amp;q&amp;f=false">Questions 659 et 660</a>, L'Intermédiaire des mathématiciens, Tome II (1895) p. 317.
%H A135535 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NewMersennePrimeConjecture.html">New Mersenne Prime Conjecture</a>
%F A135535 a(n) = 4^A059266(n) - 3. - _Ryan Propper_, Feb 26 2008
%e A135535 16381 is a term because 4^7 - 3 = 16381 is prime.
%t A135535 Do[If[PrimeQ[4^n - 3], Print[4^n - 3]], {n, 100}] (* _Robert G. Wilson v_, Feb 29 2008 *)
%t A135535 Select[4^Range[200]-3,PrimeQ] (* _Harvey P. Dale_, Jul 11 2022 *)
%Y A135535 Cf. A000040, A050415, A057732, A057733, A059266.
%K A135535 nonn
%O A135535 1,1
%A A135535 Daniele Corradetti (d.corradetti(AT)gmail.com), Feb 21 2008
%E A135535 More terms from _R. J. Mathar_, _Robert G. Wilson v_ and _Ryan Propper_, Feb 26 2008