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

1,1

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.

Links

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

Example

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.

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

Prog

(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
(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

Crossrefs

Cf. A240083.

Sequence in context: A289277 A140182 A340753 * A234943 A209494 A082910

Adjacent sequences: A239469 A239470 A239471 * A239473 A239474 A239475

Keyword

nonn

Author

Derek Orr, Mar 31 2014

Status

approved