A346154 -id:A346154 - cp's OEIS frontend

A343589 Smallest prime of the form n^k-(n-1) or 0 if no such prime exists.

3, 7, 13, 3121, 31, 43, 549755813881, 73, 991, 1321, 248821, 157, 2731, 211, 241, 34271896307617, 307, 6841, 13107199999999999999981, 421, 463, 141050039560662968926081, 331753, 601, 17551, 7625597484961, 757, 1816075630094014572464024421543167816955354437761

Offset: 2

Blake Branstool, Apr 20 2021

nonn

All values up to n=70 have been found and proved to be primes. n=71 has k=3019 and gives a probable prime.

See A113516, which gives the k values and is the main entry for these primes, for more extensively researched information. - Peter Munn, Nov 20 2021

			For n=2 and k=2, 2^2-(2-1)=3 thus a(2)=3. k is 2 as well for n=3,4.
For n=5 the first k to result in a prime is 5, 5^5-(5-1)=3121 thus a(5)=3121.

Blake Branstool, Table of n, a(n) for n = 2..70

A113516 gives the k values.

Cf. A076846, A084741, A084745, A346154.

a(n) = my(k=1, p); while (!isprime(p=n^k-(n-1)), k++); p; \\ Michel Marcus, Nov 17 2021

Name revised by Peter Munn, Nov 16 2021

A346156 Primes of the form x^k+x+1 where k >= 2 and x >= 1.

Original entry on oeis.org

3, 7, 11, 13, 19, 31, 43, 67, 73, 131, 157, 211, 223, 241, 307, 421, 463, 521, 601, 631, 733, 739, 757, 1123, 1303, 1483, 1723, 1741, 2551, 2971, 3307, 3391, 3541, 3907, 4099, 4423, 4831, 4931, 5113, 5701, 5851, 6007, 6163, 6481, 6571, 8011, 8191, 9283, 9901, 10303, 11131, 12211, 12433, 13807

Offset: 1

J. M. Bergot and Robert Israel, Jul 07 2021

nonn

Primes p such that p-1 is in A253913.
Primes with more than one representation of this form include 31 = 3^3+3+1 = 5^2+5+1 and 131 = 2^7+2+1 = 5^3+5+1.  Are there any others?
There are no others with more than one representation (except 3, trivially) < 10^19 (first 170385840 terms). - Michael S. Branicky, Jul 08 2021

			a(3) = 11 is a term because 11 = 2^3+2+1 and is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A253913, A346154.

N:= 10^8: # for terms <= N
S:= {3}:
for k from 2 to ilog2(N-1) do
  S:= S union select(t -> t<= N and isprime(t),{seq(x^k+x+1,x=2..floor(N^(1/k)))}):
od:
sort(convert(S,list));

from sympy import isprime
def aupto(lim):
    xkx = set(x**k + x + 1 for k in range(2, lim.bit_length()) for x in range(int(lim**(1/k))+2))
    return sorted(filter(isprime, filter(lambda t: t<=lim, xkx)))
print(aupto(14000)) # Michael S. Branicky, Jul 07 2021

cp's OEIS Frontend

A343589 Smallest prime of the form n^k-(n-1) or 0 if no such prime exists.

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