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Number of primes of the form k^n - 2^k for positive integers k.
4

%I #35 Feb 05 2023 09:19:00

%S 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 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 1,6,1,0,1,4,1,1,0,4,1,4,0,3,1,0,0,7,1,4

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

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

%C 1) -

%C 2) -

%C 3) 3;

%C 4) 3, 5, 7, 13;

%C 5) 9, 19, 21;

%C 6) 13, 17;

%C 7) 3, 25, 31;

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

%C 9) 13;

%C 10) 9, 23, 31, 47;

%C 11) 31, 45;

%C 12) 7, 29, 41, 47;

%C 13) -

%H Jinyuan Wang, <a href="/A245459/b245459.txt">Table of n, a(n) for n = 1..450</a>

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

%e 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:= proc(n)

%p local T,k,x;

%p T:= 0;

%p for k from 3 by 2 do

%p x:= k^n - 2^k;

%p if x <= 0 then return T fi;

%p if isprime(x) then T:= T+1 fi;

%p od:

%p end proc:

%p seq(A245459(n),n=1..100); # _Robert Israel_, Jul 23 2014

%t 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 Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Feb 05 2023, after _Robert Israel_ *)

%o (PARI)

%o a(n) = my(m=0, k=2); while(k^n>2^k, if(ispseudoprime(k^n-2^k), m++); k++); m

%o vector(80, n, a(n)) \\ _Colin Barker_, Jul 27 2014

%o (Python)

%o import sympy

%o def a(n):

%o ..k = 2

%o ..count = 0

%o ..while k**n > 2**k:

%o ....if sympy.isprime(k**n-2**k):

%o ......count += 1

%o ....k += 1

%o ..return count

%o n = 1

%o while n < 100:

%o ..print(a(n),end=', ')

%o ..n += 1 # _Derek Orr_, Aug 02 2014

%K nonn

%O 1,4

%A _Juri-Stepan Gerasimov_, Jul 22 2014

%E Terms corrected by _Robert Israel_, Jul 23 2014

%E More terms from _Colin Barker_, Jul 27 2014

%E Name edited with k range given by _Wolfdieter Lang_, Aug 15 2014

%E More terms from _Jinyuan Wang_, Feb 24 2020