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%I M1318 N0506 #76 Apr 03 2023 10:36:09
%S 1,2,5,6,8,12,18,30,36,41,66,189,201,209,276,353,408,438,534,2208,
%T 2816,3168,3189,3912,20909,34350,42294,42665,44685,48150,54792,55182,
%U 59973,80190,157169,213321,303093,362765,382449,709968,801978,916773,1832496,2145353
%N Numbers k such that 3*2^k + 1 is prime.
%C From _Zak Seidov_, Mar 08 2009: (Start)
%C List is complete up to 3941000 according to the list of RB & WK.
%C So far there are only 4 primes: 2, 5, 41, 353. (End)
%D D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 614.
%D H. Riesel, "Prime numbers and computer methods for factorization", Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see pp. 381-384.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Jeppe Stig Nielsen, <a href="/A002253/b002253.txt">Table of n, a(n) for n = 1..50</a>
%H Ray Ballinger, <a href="http://www.prothsearch.com/index.html">Proth Search Page</a>
%H Ray Ballinger and Wilfrid Keller, <a href="http://www.prothsearch.com/riesel1.html">List of primes k.2^n + 1 for k < 300</a>
%H C. K. Caldwell, <a href="http://www.utm.edu/research/primes/">The Prime Pages</a>
%H Y. Gallot, <a href="http://www.utm.edu/research/primes/programs/gallot/index.html">Proth.exe: Windows Program for Finding Large Primes</a>
%H S. W. Golomb, <a href="http://www.jstor.org/stable/2005337">Properties of the sequence 3.2^n+1</a>, Math. Comp., 30 (1976), 657-663.
%H S. W. Golomb, <a href="/A004119/a004119.pdf">Properties of the sequence 3.2^n+1</a>, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]
%H Wilfrid Keller, <a href="http://www.prothsearch.com/riesel2.html">List of primes k.2^n - 1 for k < 300</a>
%H PrimeGrid, <a href="https://www.primegrid.com/stats_321_llr.php">321 Prime Search statistics</a>
%H R. M. Robinson, <a href="/A002253/a002253_1.pdf"> A report on primes of the form k.2^n+1 and on factors of Fermat numbers</a>, Proc. Amer. Math. Soc., 9 (1958), 673-681. [Annotated scanned copy of selected pages. The first page is (accidentally) included with the scan of the Wilson letter below.]
%H R. G. Wilson v, <a href="/A002253/a002253.pdf">Letter (FAX) to N. J. A. Sloane, May 20 1994, concerning A002253-A002274, A000043, A002235-A002240, A001770-A001775, A007117, and many other sequences</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ProthPrime.html">Proth Prime</a>
%H R. G. Wilson, V, <a href="/A002982/a002982.pdf">Letter to N. J. A. Sloane, Jan. 1989</a>
%H <a href="/index/Pri#riesel">Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime</a>
%F a(n) = log_2((A039687(n)-1)/3) = floor(log_2(A039687(n)/3)). - _M. F. Hasler_, Mar 03 2023
%o (PARI) is(n)=isprime(3*2^n+1) \\ _Charles R Greathouse IV_, Feb 17 2017
%o (PARI) A2253=[1]; A002253(n)=for(k=#A2253, n-1, my(m=A2253[k]); until(ispseudoprime(3<<m+++1),); A2253=concat(A2253,m))+A2253[n] \\ _M. F. Hasler_, Mar 03 2023
%Y Cf. A002254, A004119, A181565.
%Y See A039687 for the actual primes.
%K hard,nonn
%O 1,2
%A _N. J. A. Sloane_
%E Corrected and extended according to the list of Ray Ballinger and Wilfrid Keller by _Zak Seidov_, Mar 08 2009
%E Edited by _N. J. A. Sloane_, Mar 13 2009
%E a(47) and a(48) from the Ballinger & Keller web page, _Joerg Arndt_, Apr 07 2013
%E a(49) from https://t5k.org/primes/page.php?id=116922, _Fabrice Le Foulher_, Mar 09 2014
%E Terms moved from Data to b-file (Links), and additional term appended to b-file, by _Jeppe Stig Nielsen_, Oct 30 2020