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.

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, 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, 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

Offset: 1

Author

Derek Orr, Mar 31 2014

Keywords

• nonn

Comments

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.

See A240083 for the n-values with nonzero entries.

Examples

			2^2 = 4 is not prime. 3^2-2^2 = 5 is prime. Thus, a(2) = 3.
2^3 = 8 is not prime. 3^3-2^3 = 19 is prime. Thus, a(3) = 3.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A240083.

Programs

Maple

f:= proc(n) local x, r, k;
  r:= 0; x:= 2^n;
  for k from 3 do
    r:= r + (k-1)^n;
    x:= k^n - r;
    if x < 2 then return 0 fi;
    if isprime(x) then return k fi;
  od
end proc:
f(1):= 2:
map(f, [$1..100]); # Robert Israel, Jan 03 2024

PARI

a(n)=k=1;while((s=k^n-sum(i=2,k-1,i^n))>0,if(isprime(s),return(k));k++)
for(n=1,100,print1(a(n),", ")) \\ Derek Orr, Mar 12 2015

Python

import sympy
from sympy import isprime
def Lep(n):
  for k in range(2*10**3):
    num = k**n
    for i in range(2,k):
      num -= i**n
      if num < 0:
        return None
    if isprime(num):
      return k
n = 1
while n < 100:
  if Lep(n) == None:
    print(0)
  else:
    print(Lep(n))
  n += 1