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

%I A245459 #37 May 22 2025 10:21:40
%S A245459 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,
%T A245459 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,
%U A245459 1,6,1,0,1,4,1,1,0,4,1,4,0,3,1,0,0,7,1,4
%N A245459 Number of primes of the form k^n - 2^k for positive integers k.
%C A245459 The values of k such that k^n - 2^k is prime for n = 1, 2, ..., 13 are
%C A245459    1)  -
%C A245459    2)  -
%C A245459    3)  3;
%C A245459    4)  3,  5,  7, 13;
%C A245459    5)  9, 19, 21;
%C A245459    6) 13, 17;
%C A245459    7)  3, 25, 31;
%C A245459    8)  3,  9, 13, 19, 29;
%C A245459    9) 13;
%C A245459   10)  9, 23, 31, 47;
%C A245459   11) 31, 45;
%C A245459   12)  7, 29, 41, 47;
%C A245459   13)  -
%H A245459 Jinyuan Wang, <a href="/A245459/b245459.txt">Table of n, a(n) for n = 1..450</a>
%F A245459 a(n) = |{k from positive integers: k^n - 2^k = prime}| for n >= 1. - _Wolfdieter Lang_, Aug 15 2014
%e A245459 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).
%p A245459 A245459:= proc(n)
%p A245459 local T,k,x;
%p A245459 T:= 0;
%p A245459 for k from 3 by 2 do
%p A245459   x:= k^n - 2^k;
%p A245459   if x <= 0 then return T fi;
%p A245459   if isprime(x) then T:= T+1 fi;
%p A245459 od:
%p A245459 end proc:
%p A245459 seq(A245459(n),n=1..100); # _Robert Israel_, Jul 23 2014
%t A245459 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];
%t A245459 Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Feb 05 2023, after _Robert Israel_ *)
%o A245459 (PARI)
%o A245459 a(n) = my(m=0, k=2); while(k^n>2^k, if(ispseudoprime(k^n-2^k), m++); k++); m
%o A245459 vector(80, n, a(n)) \\ _Colin Barker_, Jul 27 2014
%o A245459 (Python)
%o A245459 import sympy
%o A245459 def a(n):
%o A245459   k = 2
%o A245459   count = 0
%o A245459   while k**n > 2**k:
%o A245459     if sympy.isprime(k**n-2**k):
%o A245459       count += 1
%o A245459     k += 1
%o A245459   return count
%o A245459 n = 1
%o A245459 while n < 100:
%o A245459   print(a(n),end=', ')
%o A245459   n += 1 # _Derek Orr_, Aug 02 2014
%K A245459 nonn
%O A245459 1,4
%A A245459 _Juri-Stepan Gerasimov_, Jul 22 2014
%E A245459 Terms corrected by _Robert Israel_, Jul 23 2014
%E A245459 More terms from _Colin Barker_, Jul 27 2014
%E A245459 Name edited with k range given by _Wolfdieter Lang_, Aug 15 2014
%E A245459 More terms from _Jinyuan Wang_, Feb 24 2020