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

1,1

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)

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.

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A002383, A059054.

Cf. A003424, A085104.

Sequence in context: A342183 A206945 A206946 * A243367 A145670 A004061

Adjacent sequences: A059052 A059053 A059054 * A059056 A059057 A059058

Keyword

nonn,easy

Author

Henry Bottomley, Dec 21 2000

Status

approved