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

1,1

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)

References

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

Links

Table of n, a(n) for n=1..23.

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A000325, A081296, A052007.

Sequence in context: A175206 A032486 A123859 * A101234 A233190 A269154

Adjacent sequences: A048741 A048742 A048743 * A048745 A048746 A048747

Keyword

nonn,nice,hard,more

Author

G. L. Honaker, Jr.

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

Status

approved