A052007 Numbers m such that 2^m + m is prime.

1, 3, 5, 9, 15, 39, 75, 81, 89, 317, 701, 735, 1311, 1881, 3201, 3225, 11795, 88071, 204129, 678561

Offset: 1

Author

G. L. Honaker, Jr. and Patrick De Geest, Nov 15 1999

Keywords

• nonn
• nice
• hard
• more

Comments

Terms >= 701 are currently only strong pseudoprimes.

If m=1 (mod 6) or m=2 (mod 6) then 3 divides 2^m+m. Thus for n > 1, a(n)!=1 (mod 6) and a(n)!=2 (mod 6).

Some of the results were computed using the PrimeFormGW (PFGW) primality-testing program. - Hugo Pfoertner, Nov 14 2019

Keller (see Links) notes that a Mersenne number M(2^m+m) = 2^(2^m+m) - 1 can be written as (2^m)*2^(2^m) - 1, and lists the first twelve terms of this sequence. The last known case where M(2^m+m) is prime is for m=a(4)=9, which gives the prime M(521). - Jeppe Stig Nielsen, Apr 20 2021

Examples

			2^39 + 39 = 549755813927 is prime.
		

Links

• W. Keller, New Cullen Primes, Math. Comp. 64 (1995), 1733-1741, S39.
• Henri Lifchitz, Renaud Lifchitz, PRP Top Records. 2^n+n.

Crossrefs

Cf. A006127, A048744, A129962.

Programs

Mathematica

Do[ If[ PrimeQ[ 2^n + n ], Print[ n ] ], {n, 0, 7000} ]
v={1}; Do[If[Mod[n, 2]*(Mod[n, 6]-1)!= 0&&PrimeQ[2^n+n], v=Append[v, n]; Print[v]], {n, 2, 20000}]

PARI

is(n)=isprime(2^n+n) \\ Charles R Greathouse IV, Feb 09 2017

Extensions

11795 from Farideh Firoozbakht, Aug 21 2003

88071 from Hugo Pfoertner, Dec 26 2004

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