A059055 - OEIS

%I #57 Aug 31 2021 01:09:02

%S 3,7,11,13,31,43,61,73,157,211,241,307,421,463,521,547,601,683,757,

%T 1123,1483,1723,2551,2731,2971,3307,3541,3907,4423,4831,5113,5701,

%U 6007,6163,6481,8011,8191,9091,9901,10303,11131,12211,12433,13421,13807,14281

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

%C 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.

%C From _Bernard Schott_, Apr 30 2021: (Start)

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

%C 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)

%H Giovanni Resta, <a href="/A059055/b059055.txt">Table of n, a(n) for n = 1..10000</a> (first 3880 terms from T. D. Noe)

%H H. Dubner and T. Granlund, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.html">Primes of the Form (b^n+1)/(b+1)</a>, J. Integer Sequences, 3 (2000), #P00.2.7.

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

%t 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 *)

%o (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

%Y Cf. A002383, A059054.

%Y Cf. A003424, A085104.

%K nonn,easy

%O 1,1

%A _Henry Bottomley_, Dec 21 2000