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URL: https://oeis.org/A058013

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A058013
Smallest prime p such that (n+1)^p - n^p is prime.
9
2, 2, 2, 3, 2, 2, 7, 2, 2, 3, 2, 17, 3, 2, 2, 5, 3, 2, 5, 2, 2, 229, 2, 3, 3, 2, 3, 3, 2, 2, 5, 3, 2, 3, 2, 2, 3, 3, 2, 7, 2, 3, 37, 2, 3, 5, 58543, 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 4663, 54517, 17, 3, 2, 5, 2, 3, 3, 2, 2, 47, 61, 19, 23, 2, 2, 19, 7, 2, 7, 3, 2, 331, 2, 179, 5, 2, 5, 3, 2, 2
OFFSET
1,1
COMMENTS
The terms a(47) and a(60) [were] unknown. The sequence continues at a(48): 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 4663, a(60)=?, continued at a(61): 17, 3, 2, 5, 2, 3, 3, 2, 2, 47, 61, 19, 23, 2, 2, 19, 7, 2, 7, 3, 2, 331, 2, 179, 5, 2, 5, 3, 2, 2. - Hugo Pfoertner, Aug 27 2004
In September and November 2005, Jean-Louis Charton found a(60)=54517 and a(47)=58543, respectively. Earlier, Mike Oakes found a(106)=7639 and a(124)=5839. All these large values of a(n) yield probable primes. - T. D. Noe, Dec 05 2005, Sep 18 2008
a(106) = 6529 and a(124) = 5167 are true.
a(137) is probably 196873 from prime of this form discovered by Jean-Louis Charton in December 2009 and reported to Henri Lifchitz's PRP Top. - Robert Price, Feb 17 2012
a(138) through a(150) is 2,>32401,2,2,3,8839,5,7,2,3,5,271,13. - Robert Price, Feb 17 2012
a(276)=88301, a(139)>240000 and a(256)>100000. - Jean-Louis Charton, Jun 27 2012
Three more terms found, a(325)=81943, a(392)=64747, a(412)=56963 and also a(139)>260000, a(295)>100000, a(370)>100000, a(373)>100000. 29 unknown terms < 1000 remain. - Jean-Louis Charton, Aug 15 2012
Three more terms a(577)=55117, a(588)=60089 and a(756)=96487. - Jean-Louis Charton, Dec 13 2012
Three more (PRP) terms a(845)=83761, a(897)=48311, a(918)=54919. - Jean-Louis Charton, Dec 31 2013.
Some of the results were computed using the PrimeFormGW (PFGW) primality-testing program. - Hugo Pfoertner, Nov 14 2019
LINKS
Robert Price and Robert G. Wilson v, Table of n, a(n) for n = 1..138
Jean-Louis Charton and Robert G. Wilson v, a(n) for n=1..1000 status
FORMULA
a((p-1)/2) = 2 for odd primes p. - Alexander Adamchuk, Dec 01 2006
MATHEMATICA
lmt = 10000; f[n_] := Block[{p = 2}, While[p < lmt && !PrimeQ[(n+1)^p - n^p], p = NextPrime@ p]; If[p > lmt, 0, p]]; Do[ Print[{n, f[n] // Timing}], {n, 1000}] (* Robert G. Wilson v, Dec 01 2014 *)
spp[n_]:=Module[{p=2}, While[!PrimeQ[(n+1)^p-n^p], p=NextPrime[p]]; p]; Array[spp, 90] (* Harvey P. Dale, Jul 01 2025 *)
PROG
(PARI) a(n)=forprime(p=2, default(primelimit), if(ispseudoprime((n+1)^p-n^p), return(p))) \\ Charles R Greathouse IV, Feb 20 2012
KEYWORD
nonn,nice
AUTHOR
Robert G. Wilson v, Nov 13 2000
EXTENSIONS
More terms from T. D. Noe, Dec 05 2005
Typo in first Mathematica program corrected by Ray Chandler, Feb 22 2017
STATUS
approved