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A239472 Least number k such that k^n-(k-1)^n-...-3^n-2^n is prime. a(n) = 0 if no such number exists.

%I A239472 #40 May 22 2025 10:21:37
%S A239472 2,3,3,7,3,0,0,0,0,7,7,0,4,0,8,11,3,16,15,0,4,7,0,23,0,19,12,11,3,0,3,
%T A239472 7,12,0,12,0,0,0,0,0,16,0,0,0,59,11,44,32,16,0,0,0,3,0,23,0,20,75,3,0,
%U A239472 28,0,0,0,36,0,60,0,0,0,36,0,0,0,0,19,0,0,0,0,0,91,75,0,0,0,32,108,7,0,60,0,40,39,0,0,0,0,80
%N A239472 Least number k such that k^n-(k-1)^n-...-3^n-2^n is prime. a(n) = 0 if no such number exists.
%C A239472 a(n) = 0 for n = {6, 7, 8, 9, 12, 14, 20, 23, 25, ...} because for k large enough, k^n-(k-1)^n-...-3^n-2^n < 0. Thus, no number will be prime.
%C A239472 See A240083 for the n-values with nonzero entries.
%H A239472 Robert Israel, <a href="/A239472/b239472.txt">Table of n, a(n) for n = 1..1000</a>
%e A239472 2^2 = 4 is not prime. 3^2-2^2 = 5 is prime. Thus, a(2) = 3.
%e A239472 2^3 = 8 is not prime. 3^3-2^3 = 19 is prime. Thus, a(3) = 3.
%p A239472 f:= proc(n) local x, r, k;
%p A239472   r:= 0; x:= 2^n;
%p A239472   for k from 3 do
%p A239472     r:= r + (k-1)^n;
%p A239472     x:= k^n - r;
%p A239472     if x < 2 then return 0 fi;
%p A239472     if isprime(x) then return k fi;
%p A239472   od
%p A239472 end proc:
%p A239472 f(1):= 2:
%p A239472 map(f, [$1..100]); # _Robert Israel_, Jan 03 2024
%o A239472 (Python)
%o A239472 import sympy
%o A239472 from sympy import isprime
%o A239472 def Lep(n):
%o A239472   for k in range(2*10**3):
%o A239472     num = k**n
%o A239472     for i in range(2,k):
%o A239472       num -= i**n
%o A239472       if num < 0:
%o A239472         return None
%o A239472     if isprime(num):
%o A239472       return k
%o A239472 n = 1
%o A239472 while n < 100:
%o A239472   if Lep(n) == None:
%o A239472     print(0)
%o A239472   else:
%o A239472     print(Lep(n))
%o A239472   n += 1
%o A239472 (PARI) a(n)=k=1;while((s=k^n-sum(i=2,k-1,i^n))>0,if(isprime(s),return(k));k++)
%o A239472 for(n=1,100,print1(a(n),", ")) \\ _Derek Orr_, Mar 12 2015
%Y A239472 Cf. A240083.
%K A239472 nonn
%O A239472 1,1
%A A239472 _Derek Orr_, Mar 31 2014