A243160 Least number k such that k^n - k^(n-1) + k^(n-2) - ... + (-1)^n is prime or 0 if no such k exists.

3, 2, 2, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 7, 0, 2, 0, 0, 0, 0, 0, 16, 0, 0, 0, 61, 0, 2, 0, 0, 0, 6, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 0, 0, 0, 5, 0, 0, 0, 46, 0, 18, 0, 0, 0, 0, 0, 2, 0, 0, 0, 49, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 70

Offset

1,1

Comments

If n is in A006093, then a(n) is not 0.

That is only a conjecture. - Robert Israel, Apr 15 2026

The only term where the converse is false is a(3) = 2. Also, 2 is the only such number that makes k^3 - k^2 + k - 1 prime. Otherwise, a(n) is not 0 iff n is in A006093.

Links

Table of n, a(n) for n=1..96.

Example

1^4 - 1^3 + 1^2 - 1^1 + 1 = 1 is not prime. 2^4 - 2^3 + 2^2 - 2^1 + 1 = 11 is prime. Thus a(4) = 2.

Maple

f:= proc(n) local j, P, x, k;
  if not isprime(n+1) then return 0 fi;
  P:= add(x^(n-j)*(-1)^j, j=0..n);
  for k from 1 do if isprime(eval(P, x=k)) then return k fi od;
end proc:
f(3):= 2:
map(f, [$1..100]); # Robert Israel, Apr 16 2026

Prog

(PARI) a(n)=for(k=1, 1000, s=k^n; if(ispseudoprime(s+sum(i=1, n, (-1)^i*k^(n-i))), return(k)))
n=1; while(n<100, print1(a(n), ", "); n+=1)
(PARI) A243160(n) = if(3!=n && !isprime(1+n), 0, my(s); for(k=1, oo, s=k^n; if(isprime(s+sum(i=1, n, (-1)^i*k^(n-i))), return(k)))); \\ Antti Karttunen, Jul 19 2024

Crossrefs

Cf. A006093.

Sequence in context: A098054 A336987 A075801 * A272694 A292370 A389963

Adjacent sequences: A243157 A243158 A243159 * A243161 A243162 A243163

Keyword

nonn

Author

Derek Orr, May 31 2014

Status

approved