A059055 Primes which can be written as (b^k+1)/(b+1) for positive integers b and k.

3, 7, 11, 13, 31, 43, 61, 73, 157, 211, 241, 307, 421, 463, 521, 547, 601, 683, 757, 1123, 1483, 1723, 2551, 2731, 2971, 3307, 3541, 3907, 4423, 4831, 5113, 5701, 6007, 6163, 6481, 8011, 8191, 9091, 9901, 10303, 11131, 12211, 12433, 13421, 13807, 14281

Offset: 1

Author

Henry Bottomley, Dec 21 2000

Keywords

• nonn
• easy

Comments

For (b^k+1)/(b+1) to be a prime, k must be an odd prime. 2=(0^0+1)/(0+1) has been excluded since neither b nor k would be positive.

From Bernard Schott, Apr 30 2021: (Start)

43 is the only known prime to have two such representations (examples).

The next two sequences realize a partition of this set: Brazilian primes of the form (c^q-1)/(c-1) (A002383 \ {3}) and primes that are not Brazilian (A343774). (End)

Examples

			43 is in the sequence since (2^7+1)/(2+1) = 129/3 = 43; indeed also (7^3+1)/(7+1) = 344/8 = 43.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 3880 terms from T. D. Noe)
• H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.

Crossrefs

Cf. A002383, A059054.

Cf. A003424, A085104.

Programs

Mathematica

max = 89; maxdata = (1 + max^3)/(1 + max); a = {}; Do[i = 1; While[i = i + 2; cc = (1 + m^i)/(1 + m); cc <= maxdata, If[PrimeQ[cc], a = Append[a, cc]]], {m, 2, max}]; Union[a] (* Lei Zhou, Feb 08 2012 *)

PARI

isok(p) = {if (isprime(p), for (b=2, p, my(k=3); while ((x=(b^k+1)/(b+1)) <= p, if (x == p, return (1)); k = nextprime(k+1););););} \\ Michel Marcus, Apr 30 2021