A240081 Least number k such that k^n-(k-1)^n-...-3^n-2^n-1 is prime.

0, 2, 2, 5, 2, 6, 2, 5, 0, 0, 0, 6, 2, 0, 0, 6, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0

Offset

1,2

Comments

a(n) = 0 for n = {1, 9, 10, 11, 14, 15, 18...} because at certain k-values, k^n-(k-1)^n-...-3^n-2^n-1 becomes a strictly decreasing negative sequence. Thus, no number will be prime.

See A240082 for the n-values with nonzero entries.

Links

Giovanni Resta, Table of n, a(n) for n = 1..6000

Example

2^4-1 = 15 is not prime. 3^4-2^4-1 = 64 is not prime. 4^4-3^4-2^4-1 = 158 is not prime. 5^4-4^4-3^4-2^4-1 = 271 is prime. Thus, a(4) = 5.

Mathematica

a[n_] := Block[{s=1, k=1}, While[s > 0 && ! PrimeQ[s], s -= 2*k^n; s += (++k)^n]; If[s > 0, k, 0]]; Array[a, 90] (* Giovanni Resta, Apr 01 2014 *)

Prog

(Python)
import sympy
from sympy import isprime
def Leq(n):
  for k in range(1000):
    num = k**n
    for i in range(2, k):
      num -= i**n
      if num < 1:
        return None
    if isprime(num-1):
      return k
n = 1
while n < 100:
  if Leq(n) == None:
    print(0)
  else:
    print(Leq(n))
  n += 1

Crossrefs

Cf. A240082.

Sequence in context: A319895 A323504 A011143 * A305791 A382524 A299764

Adjacent sequences: A240078 A240079 A240080 * A240082 A240083 A240084

Keyword

nonn

Author

Derek Orr, Mar 31 2014

Status

approved