A245459 Number of primes of the form k^n - 2^k for positive integers k.

0, 0, 1, 4, 3, 2, 3, 5, 1, 4, 2, 4, 0, 6, 2, 2, 1, 2, 3, 5, 1, 9, 1, 4, 2, 3, 1, 2, 2, 2, 1, 4, 1, 5, 1, 2, 3, 3, 1, 2, 2, 1, 0, 3, 0, 1, 1, 2, 1, 4, 0, 1, 0, 3, 0, 3, 0, 2, 1, 4, 5, 3, 0, 3, 5, 9, 1, 5, 1, 6, 1, 0, 1, 4, 1, 1, 0, 4, 1, 4, 0, 3, 1, 0, 0, 7, 1, 4

Offset: 1

Author

Juri-Stepan Gerasimov, Jul 22 2014

Keywords

• nonn

Comments

The values of k such that k^n - 2^k is prime for n = 1, 2, ..., 13 are

1) -

2) -

3) 3;

4) 3, 5, 7, 13;

5) 9, 19, 21;

6) 13, 17;

7) 3, 25, 31;

8) 3, 9, 13, 19, 29;

9) 13;

10) 9, 23, 31, 47;

11) 31, 45;

12) 7, 29, 41, 47;

13) -

Examples

			a(4) = 4 because 3^4 - 2^3 = 73 (prime), 5^4 - 2^5 = 593 (prime), 7^4 - 2^7 = 2273 (prime), 13^4 - 2^13 = 20369 (prime).
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..450

Programs

Maple

A245459:= proc(n)
local T,k,x;
T:= 0;
for k from 3 by 2 do
  x:= k^n - 2^k;
  if x <= 0 then return T fi;
  if isprime(x) then T:= T+1 fi;
od:
end proc:
seq(A245459(n),n=1..100); # Robert Israel, Jul 23 2014

Mathematica

a[n_] := Module[{cnt = 0, k, x}, For[k = 3, True, k = k+2, x = k^n-2^k; If[x <= 0, Return[cnt]]; If[PrimeQ[x], cnt++]]; cnt];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 05 2023, after Robert Israel *)

PARI

a(n) = my(m=0, k=2); while(k^n>2^k, if(ispseudoprime(k^n-2^k), m++); k++); m
vector(80, n, a(n)) \\ Colin Barker, Jul 27 2014

Python

import sympy
def a(n):
  k = 2
  count = 0
  while k**n > 2**k:
    if sympy.isprime(k**n-2**k):
      count += 1
    k += 1
  return count
n = 1
while n < 100:
  print(a(n),end=', ')
  n += 1 # Derek Orr, Aug 02 2014

Formula

a(n) = |{k from positive integers: k^n - 2^k = prime}| for n >= 1. - Wolfdieter Lang, Aug 15 2014

Extensions

Terms corrected by Robert Israel, Jul 23 2014

More terms from Colin Barker, Jul 27 2014

Name edited with k range given by Wolfdieter Lang, Aug 15 2014

More terms from Jinyuan Wang, Feb 24 2020