A050414 Numbers k such that 2^k - 3 is prime.

3, 4, 5, 6, 9, 10, 12, 14, 20, 22, 24, 29, 94, 116, 122, 150, 174, 213, 221, 233, 266, 336, 452, 545, 689, 694, 850, 1736, 2321, 3237, 3954, 5630, 6756, 8770, 10572, 14114, 14400, 16460, 16680, 20757, 26350, 30041, 34452, 36552, 42689, 44629, 50474, 66422, 69337, 116926, 119324, 123297, 189110, 241004, 247165, 284133, 354946, 394034, 702194, 750740, 840797, 1126380, 1215889, 1347744, 1762004, 2086750

Offset: 1

Author

Jud McCranie, Dec 22 1999

Keywords

• nonn

Comments

With 65 known primes corresponding to k < 1762005, these primes appear to be more common than Mersenne primes. Of course at this time, the larger terms correspond only to probable primes. - Paul Bourdelais, Feb 04 2012

The numbers 2^k-3 and 2^k-1 are both primes for k = 3, 5, ? The lesser number 2^p-3 is prime for primes p = 3, 5, 29, 233, 42689, 69337, ... (see A283266). - Thomas Ordowski, Sep 18 2015

The terms a(43)-a(49) were found by Paul Underwood, a(50)-a(51) found by M. Frind and P. Underwood, a(52) found by Gary Barnes, a(53)-a(58) found by M. Frind and P. Underwood, and a(59)-a(66) found by Paul Bourdelais (see link Henri Lifchitz and Renaud Lifchitz). - Elmo R. Oliveira, Dec 02 2023

Examples

			k = 22, 2^22 - 3 = 4194301 is prime.
k = 24, 2^24 - 3 = 16777213 is prime.
		

Links

• Keith Conrad, Square patterns and infinitude of primes, University of Connecticut, 2019.
• Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n-3, PRP Top Records.

Crossrefs

Cf. A045768, A050415, A057732 (numbers k such that 2^k + 3 is prime).

For prime terms see A283266.

Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), A059608 (d=5), A059609 (d=7), A059610 (d=9), A096817 (d=11), A096818 (d=13), A059612 (d=15), A059611 (d=17), A096819 (d=19), A096820 (d=21), A057220 (d=23), A356826 (d=29).

Programs

Mathematica

Do[ If[ PrimeQ[ 2^n -3 ], Print[n]], { n, 1, 15000 }]

PARI

for(n=2, 10^5, if(ispseudoprime(2^n-3), print1(n, ", "))) \\ Felix Fröhlich, Jun 23 2014

Extensions

More terms from Robert G. Wilson v, Sep 15 2000

More terms from Andrey V. Kulsha, Feb 11 2001

a(40) verified with 20 iterations of Miller-Rabin test, from Dmitry Kamenetsky, Jul 12 2008

a(41) a new PRP term, from Serge Batalov, Oct 20 2008

Corrected and extended by including two smaller (apparently known) PRP and 16 larger terms from PRP Top Records of this form, all discovered by M. Frind & P. Underwood, Gary Barnes, Oct 20 2008

a(59)-a(60) discovered by Paul Bourdelais, Mar 26 2012

a(61)-a(63) discovered by Paul Bourdelais, Jun 18 2019

a(64) discovered by Paul Bourdelais, Jul 16 2019

a(65) discovered by Paul Bourdelais, Apr 20 2020

a(66) discovered by Paul Bourdelais, May 28 2020