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A059055
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Primes which can be written as (b^k+1)/(b+1) for positive integers b and k.
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11
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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
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OFFSET
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1,1
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COMMENTS
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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.
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)
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LINKS
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EXAMPLE
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43 is in the sequence since (2^7+1)/(2+1) = 129/3 = 43; indeed also (7^3+1)/(7+1) = 344/8 = 43.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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