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A058013 Smallest prime p such that (n+1)^p - n^p is prime. 9

%I

%S 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,

%T 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,

%U 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

%N Smallest prime p such that (n+1)^p - n^p is prime.

%C 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

%C 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

%C a(106) = 6529 and a(124) = 5167 are true.

%C 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

%C a(138) through a(150) is 2,>32401,2,2,3,8839,5,7,2,3,5,271,13. - _Robert Price_, Feb 17 2012

%C a(276)=88301, a(139)>240000 and a(256)>100000. - _Jean-Louis Charton_, Jun 27 2012

%C 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

%C Three more terms a(577)=55117, a(588)=60089 and a(756)=96487. - _Jean-Louis Charton_, Dec 13 2012

%C Three more (PRP) terms a(845)=83761, a(897)=48311, a(918)=54919. - _Jean-Louis Charton_, Dec 2012-2013.

%H Robert Price and Robert G. Wilson v, <a href="/A058013/b058013.txt">Table of n, a(n) for n = 1..138</a>

%H Jean-Louis Charton and Robert G. Wilson v, <a href="/A058013/a058013_4.txt">a(n) for n=1..1000 status</a>

%H Richard Fischer, <a href="http://www.fermatquotient.com/PrimSerien/PrimPot.txt">Generalized primes of the form (B+1)^N - B^N.</a>

%H Yahoo Groups, <a href="http://groups.yahoo.com/group/primeform/">User group for PFGW & PrimeForm programs.</a>

%F a((p-1)/2) = 2 for odd primes p. - _Alexander Adamchuk_, Dec 01 2006

%t 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 *)

%o (PARI) a(n)=forprime(p=2,default(primelimit),if(ispseudoprime((n+1)^p-n^p),return(p))) \\ _Charles R Greathouse IV_, Feb 20 2012

%Y Cf. A065913, A103794, A115596, A247244.

%Y Cf. A000043, A057468, A059801, A059802, A062572-A062666, A215538.

%K nonn,nice

%O 1,1

%A _Robert G. Wilson v_, Nov 13 2000

%E More terms from _T. D. Noe_, Dec 05 2005

%E Typo in Mathematica program corrected by _Ray Chandler_, Feb 22 2017

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Last modified June 18 18:52 EDT 2019. Contains 324215 sequences. (Running on oeis4.)