A002253 Numbers k such that 3*2^k + 1 is prime. (Formerly M1318 N0506)

1, 2, 5, 6, 8, 12, 18, 30, 36, 41, 66, 189, 201, 209, 276, 353, 408, 438, 534, 2208, 2816, 3168, 3189, 3912, 20909, 34350, 42294, 42665, 44685, 48150, 54792, 55182, 59973, 80190, 157169, 213321, 303093, 362765, 382449, 709968, 801978, 916773, 1832496, 2145353

Offset

1,2

Comments

From Zak Seidov, Mar 08 2009: (Start)

List is complete up to 3941000 according to the list of RB & WK.

So far there are only 4 primes: 2, 5, 41, 353. (End)

References

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 614.

H. Riesel, "Prime numbers and computer methods for factorization", Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see pp. 381-384.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..50

Ray Ballinger, Proth Search Page

Ray Ballinger and Wilfrid Keller, List of primes k.2^n + 1 for k < 300

C. K. Caldwell, The Prime Pages

Y. Gallot, Proth.exe: Windows Program for Finding Large Primes

S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.

S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]

Wilfrid Keller, List of primes k.2^n - 1 for k < 300

PrimeGrid, 321 Prime Search statistics

R. M. Robinson, A report on primes of the form k.2^n+1 and on factors of Fermat numbers, 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.]

R. G. Wilson v, Letter (FAX) to N. J. A. Sloane, May 20 1994, concerning A002253-A002274, A000043, A002235-A002240, A001770-A001775, A007117, and many other sequences

Eric Weisstein's World of Mathematics, Proth Prime

R. G. Wilson, V, Letter to N. J. A. Sloane, Jan. 1989

Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Formula

a(n) = log_2((A039687(n)-1)/3) = floor(log_2(A039687(n)/3)). - M. F. Hasler, Mar 03 2023

Prog

(PARI) is(n)=isprime(3*2^n+1) \\ Charles R Greathouse IV, Feb 17 2017
(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

Crossrefs

Cf. A002254, A004119, A181565.

See A039687 for the actual primes.

Sequence in context: A280381 A333965 A105107 * A032716 A132229 A028750

Adjacent sequences: A002250 A002251 A002252 * A002254 A002255 A002256

Keyword

hard,nonn

Author

N. J. A. Sloane

Extensions

Corrected and extended according to the list of Ray Ballinger and Wilfrid Keller by Zak Seidov, Mar 08 2009

Edited by N. J. A. Sloane, Mar 13 2009

a(47) and a(48) from the Ballinger & Keller web page, Joerg Arndt, Apr 07 2013

a(49) from https://t5k.org/primes/page.php?id=116922, Fabrice Le Foulher, Mar 09 2014

Terms moved from Data to b-file (Links), and additional term appended to b-file, by Jeppe Stig Nielsen, Oct 30 2020

Status

approved