A247244 -id:A247244 - cp's OEIS frontend

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

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

Robert G. Wilson v, Nov 13 2000

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

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
Richard Fischer, Generalized primes of the form (B+1)^N - B^N.

Cf. A065913, A103794, A115596, A247244.

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

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 *)

a(n)=forprime(p=2,default(primelimit),if(ispseudoprime((n+1)^p-n^p),return(p))) \\ Charles R Greathouse IV, Feb 20 2012

a((p-1)/2) = 2 for odd primes p. - Alexander Adamchuk, Dec 01 2006

More terms from T. D. Noe, Dec 05 2005

Typo in first Mathematica program corrected by Ray Chandler, Feb 22 2017

A301510 Smallest positive number b such that ((b+1)^prime(n) + b^prime(n))/(2*b + 1) is prime, or 0 if no such b exists.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 3, 16, 1, 11, 6, 37, 1, 9, 120, 9, 1, 2, 67, 16, 1, 26, 103, 12, 60, 1, 239, 4, 40, 2, 44, 174, 33, 1, 3, 260, 114, 1, 161, 70, 1, 3, 2, 3, 50, 45, 472, 228, 183, 66, 37, 7, 122, 235, 68, 102, 294, 8, 13, 1, 40, 62, 143, 1, 61, 7

Offset: 2

Tim Johannes Ohrtmann, Mar 22 2018

nonn

Conjecture: a(n) > 0 for every n > 1.

Records: 1, 4, 16, 37, 120, 239, 260, 472, 917, 1539, 6633, 7050, 12818, ..., which occur at n = 2, 10, 13, 17, 20, 32, 41, 52, 72, 128, 171, 290, 309, ... - Robert G. Wilson v, Jun 16 2018

			a(10) = 4 because (5^29 + 4^29)/9 = 2149818248341 is prime and (2^29 + 1^29)/3, (3^29 + 2^29)/5 and (4^29 + 3^29)/7 are all composite.

Robert G. Wilson v, Table of n, a(n) for n = 2..345
Richard Fischer, Primzahlen mit der Form [(B+1)^N+B^N]/(2*B+1)
Henri Lifchitz & Renaud Lifchitz, Search for: (a^n+b^n)/c

Numbers n such that ((b+1)^n + b^n)/(2*b + 1) is prime for b = 1 to 18: A000978, A057469, A128066, A128335, A128336, A187805, A181141, A187819, A217095, A185239, A213216, A225097, A224984, A221637, A227170, A228573, A227171, A225818.

Cf. A058013, A103794, A222119, A247244, A250201.

Table[p = Prime[n]; k = 1; While[q = ((b+1)^n+b^n)/(2*b+1); ! PrimeQ[q], k++]; k, {n, 200}]
f[n_] := Block[{b = 1, p = Prime@ n}, While[! PrimeQ[((b +1)^p + b^p)/(2b +1)], b++]; b]; Array[f, 70, 2] (* Robert G. Wilson v, Jun 13 2018 *)

for(n=2, 200, b=0; until(isprime((((b+1)^prime(n)+b^prime(n))/(2*b+1))), b++); print1(b,", ")) \\ corrected by Eric Chen, Jun 06 2018

a(n) = A250201(2*prime(n)) - 1 for n >= 2. - Eric Chen, Jun 06 2018

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

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