A048744 Numbers k such that 2^k - k is prime.

2, 3, 9, 13, 19, 21, 55, 261, 3415, 4185, 7353, 12213, 44169, 60975, 61011, 108049, 182451, 228271, 481801, 500899, 505431, 1015321, 1061095

Offset: 1

Author

G. L. Honaker, Jr.

Keywords

• nonn
• nice
• hard
• more

Comments

All terms except for the first are odd. - Joerg Arndt, Jul 19 2016

From Iain Fox, Nov 14 2017: (Start)

If k is congruent to 5 mod 6, then 3 divides 2^k - k; therefore a(n) is never congruent to 5 mod 6.

For even k, 2^k - k is divisible by 2; thus all terms other than 2 are odd.

It follows that for n > 1, a(n) is congruent to {1, 3} mod 6.

(End)

Examples

			2^55 - 55 = 36028797018963913 is prime, so 55 is a term.
		

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 261, p. 70, Ellipses, Paris 2008.

Links

• Henri Lifchitz, Renaud Lifchitz, PRP Top Records. 2^n-n.

Crossrefs

Cf. A000325, A081296, A052007.

Programs

Mathematica

Do[ If[ PrimeQ[ 2^n - n ], Print[ n ] ], {n, 0, 7353} ]
(* Second program: *)
Select[Range[8000], PrimeQ[2^# - #] &] (* Michael De Vlieger, Nov 15 2017 *)

PARI

for(n=1,10^5,if(ispseudoprime(2^n-n),print1(n,", "))) \\ Derek Orr, Sep 01 2014

Extensions

261 and 3415 found by Warut Roonguthai

4185 and 7353 are probable primes (the latter was found by Jud McCranie).

12213 found by Robert G. Wilson v, Jan 02 2001

More terms from Henri Lifchitz contributed by Ray Chandler, Mar 02 2007

Edited by T. D. Noe, Oct 30 2008

a(22)-a(23) from Henri Lifchitz contributed by Robert Price, Sep 01 2014