

A050414


Numbers k such that 2^k  3 is prime.


42



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


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^k3 and 2^k1 are both primes for k = 3, 5, ? The lesser number 2^p3 is prime for primes p = 3, 5, 29, 233, 42689, 69337, ...  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


LINKS

Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n3, PRP Top Records.


EXAMPLE

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


MATHEMATICA

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


PROG

(PARI) for(n=2, 10^5, if(ispseudoprime(2^n3), print1(n, ", "))) \\ Felix Fröhlich, Jun 23 2014


CROSSREFS

Cf. 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).


KEYWORD

nonn


AUTHOR



EXTENSIONS

a(40) verified with 20 iterations of MillerRabin test, from Dmitry Kamenetsky, Jul 12 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


STATUS

approved



