A002253 Numbers k such that 3*2^k + 1 is prime.

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

Author

N. J. A. Sloane

Keywords

• hard
• nonn

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

Crossrefs

Cf. A002254, A004119, A181565.

See A039687 for the actual primes.

Programs

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. F. Hasler, Mar 03 2023

Formula

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

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