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

1,4

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) -

Links

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

Formula

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

Example

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).

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 *)

Prog

(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

Crossrefs

Sequence in context: A076576 A332558 A183197 * A001368 A197224 A340739

Adjacent sequences: A245456 A245457 A245458 * A245460 A245461 A245462

Keyword

nonn

Author

Juri-Stepan Gerasimov, Jul 22 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

Status

approved