A097764 -id:A097764 - cp's OEIS frontend

A239474 Smallest k >= 1 such that k^n-n is prime. a(n) = 0 if no such k exists.

3, 2, 2, 0, 4, 5, 60, 3, 2, 21, 28, 5, 2, 199, 28, 0, 234, 11, 2, 3, 2, 159, 10, 31, 68, 145, 0, 69, 186, 163, 32, 253, 26, 261, 4, 0, 8, 11, 62, 3, 22, 43, 6, 7, 8, 945, 76, 7, 116, 129, 382, 93, 330, 361, 2, 555, 224, 1359, 78, 29, 62, 39, 110, 0, 1032, 37, 462, 29

Offset: 1

Derek Orr, Mar 20 2014

nonn

If n is of the form (pk)^p for some k and some prime p, then a(n) = 0 (See A097764).

			1^1-1 = 0 is not prime. 2^1-1 = 1 is not prime. 3^1-1 = 2 is prime. Thus, a(1) = 3.

Cf. A072883, A028870, A153974, A239413, A239414, A239415, A239416, A239417, A239418.

import sympy
from sympy import isprime
def TwoMin(x):
  for k in range(1,5000):
    if isprime(k**x-x):
      return k
x = 1
while x < 100:
  print(TwoMin(x))
  x += 1

a(A097764(n)) = 0 for all n.

A241427 Smallest prime of the form n^k - k^n for some k, or 0 if no such prime exists.

Original entry on oeis.org

7, 2, 3, 6102977801, 5, 79792265017612001, 7, 2486784401

Offset: 2

Derek Orr, Aug 08 2014

nonn

Conjecture: a(n) > 0 for all n not in A097764.
More terms in b-file. If n > 4 and in A097764, n^k - k^n is factorable and won't be prime.
a(17) > 17^7500 - 7500^17. See A239279.

Derek Orr, Table of n, a(n) for n = 2..16

Cf. A078201.

a(n)=k=1;if(n>4,forprime(p=1,100,if(ispower(n)&&ispower(n)%p==0&&n%p==0,return(0));if(n%p==n,break)));k=1;while(!ispseudoprime(n^k-k^n),k++);return(n^k-k^n)
vector(15, n, a(n+1))

A097794 Least k such that the absolute value of k^n-n is prime or zero if no such k exists.

Original entry on oeis.org

3, 2, 1, 1, 4, 1, 60, 1, 2, 21, 28, 1, 2, 1, 28, 0, 234, 1, 2, 1, 2, 159, 10, 1, 68, 145, 0, 69, 186, 1, 32, 1, 26, 261, 4, 0, 8, 1, 62, 3, 22, 1, 6, 1, 8, 945, 76, 1, 116, 129, 382, 93, 330, 1, 2, 555, 224, 1359, 78, 1, 62, 1, 110, 0, 1032, 37, 462, 1, 100, 9, 88, 1, 1416, 1, 218

Offset: 1

T. D. Noe, Aug 24 2004

nonn

Because the polynomial x^n - n is reducible for n in A097764, a(n) is 0 for n=16, 27, 36, 64, 100,.... Although x^4-4 is reducible, the factor x^2-2 is -1 for x=1.

Cf. A097764 (n such that x^n-n is reducible), A072883 (least k such that k^n+n is prime).

Table[If[MemberQ[{16, 27, 36, 64, 100}, n], 0, k=1; While[ !PrimeQ[k^n-n], k++ ]; k], {n, 100}]

A239475 Least number k such that k^n + n and k^n - n are both prime, or 0 if no such number exists.

Original entry on oeis.org

4, 3, 2, 0, 42, 175, 66, 3, 2, 4983, 1770, 55055, 28686, 18765, 8456, 0, 594, 128345, 136080, 81, 92, 1163409, 18810, 10415, 11754, 3855, 0, 86043, 38880, 17639, 26088, 37293, 5540, 612015, 6876, 0, 44220, 130425, 110, 9292527, 1004850, 1812149, 442404, 1007445, 570658

Offset: 1

Derek Orr, Mar 20 2014

nonn

a(n) = 0 iff n is of the form (pk)^p for some k and some prime p (See A097764).

gcd(n,a(n)) = 1 for all a(n) > 0.

			1^1 +/- 1 = 2 and 0 are not both primes. 2^1 +/- 1 = 3 and 1 are not both primes. 3^1 +/- 1 = 4 and 2 are not both primes. 4^1 +/- 1 = 5 and 3 are both primes. Thus a(1) = 4.

Cf. A072883, A028870, A153974, A239413, A239414, A239415, A239416, A239417, A239418, A239474.

a(n)=for(k=1,10^7,if(ispseudoprime(k^n-n)&&ispseudoprime(k^n+n),return(k)))
n=1;while(n<100,print1(a(n),", ");n++)

import sympy
from sympy import isprime
def TwoBoth(x):
  for k in range(1,10**7):
    if isprime(k**x+x) and isprime(k**x-x):
      return k
x = 1
while x < 100:
  if TwoBoth(x) != None:
    print(TwoBoth(x))
  else:
    print(0)
  x += 1

a(A097764(n)) = 0 for all n.

cp's OEIS Frontend

A239474 Smallest k >= 1 such that k^n-n is prime. a(n) = 0 if no such k exists.

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A241427 Smallest prime of the form n^k - k^n for some k, or 0 if no such prime exists.

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A097794 Least k such that the absolute value of k^n-n is prime or zero if no such k exists.

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Mathematica

A239475 Least number k such that k^n + n and k^n - n are both prime, or 0 if no such number exists.

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Python

Formula