A222119 -id:A222119 - cp's OEIS frontend

A103794 Smallest number b such that b^prime(n) - (b-1)^prime(n) is prime.

2, 2, 2, 2, 6, 2, 2, 2, 6, 3, 2, 40, 7, 5, 13, 3, 3, 2, 7, 18, 47, 8, 6, 2, 26, 3, 42, 2, 13, 8, 2, 8, 328, 8, 9, 45, 27, 13, 76, 15, 52, 111, 5, 15, 50, 287, 16, 5, 40, 23, 110, 368, 23, 68, 28, 96, 81, 150, 3, 143, 4, 12, 403, 4, 45, 11, 83, 21, 96, 5, 109, 350, 128, 304, 38, 4, 163

Offset: 1

Lei Zhou, Feb 24 2005

nonn

Conjecture: sequence is defined for all positive indices.

For p=prime(n), Eisenstein's irreducibility criterion can be used to show that the polynomial (x+1)^p-x^p is irreducible, which is a necessary (but not sufficient) condition for a(n) to exist. - T. D. Noe, Dec 05 2005

			2^prime(1)-1^prime(1)=3 is prime, so a(1)=2;
2^prime(5)-1^prime(5)=2047 has a factor of 23;
...
6^prime(5)-5^prime(5)=313968931 is prime, so a(5)=6;

Cf. A103795, A066180, A058013, A222119.

f:= proc(n) local p,b;
  p:= ithprime(n);
  for b from 2 do
    if isprime(b^p - (b-1)^p) then return b fi
  od
end proc:
map(f, [$1..80]); # Robert Israel, Jun 04 2024

Do[p=Prime[k]; n=2; nm1=n-1; cp=n^p-nm1^p; While[ !PrimeQ[cp], n=n+1; nm1=n-1; cp=n^p-nm1^p]; Print[n], {k, 1, 200}]

a(n) = A222119(n) + 1. - Ray Chandler, Feb 26 2017

A222120 Number of digits in the smallest prime of the form (k+1)^p - k^p, where p = prime(n).

Original entry on oeis.org

1, 1, 2, 3, 9, 4, 6, 6, 18, 14, 10, 60, 35, 31, 53, 26, 29, 19, 57, 90, 122, 72, 65, 27, 138, 49, 168, 33, 122, 103, 39, 119, 345, 126, 143, 250, 225, 182, 315, 204, 308, 371, 134, 227, 335, 489, 255, 156, 364, 312, 476, 613, 329, 460, 372, 522, 514, 590, 133

Offset: 1

Vladimir Pletser, Feb 07 2013

The smallest primes of the form (k+1)^p - k^p are in A121620. The values of k are in A222119. For the primes p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, ... (A000043), k = 1 and Mersenne primes 2^p - 1 (A000668) are obtained.

			a(5) = 9 because the 5th prime is 11, and the smallest prime of the form (k+1)^11 - k^11 is 6^11 - 5^11 = 313968931, which has 9 digits

Cf. A222119 (values of k).

Table[p = Prime[n]; k = 1; While[q = (k + 1)^p - k^p; ! PrimeQ[q], k++]; Length[IntegerDigits[q]], {n, 60}] (* T. D. Noe, Feb 12 2013 *)

a222120(n) = {local(p,k); p=prime(n); while(!isprime((k+1)^p - k^p), k=k+1); ceil(log((k+1)^p - k^p)/log(10))} \\ Michael B. Porter, Feb 12 2013

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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A103794 Smallest number b such that b^prime(n) - (b-1)^prime(n) is prime.

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A222120 Number of digits in the smallest prime of the form (k+1)^p - k^p, where p = prime(n).

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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.

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