A365196 a(n) is the least k such that 2^k + n is not squarefree.

2, 3, 1, 0, 2, 2, 1, 0, 0, 4, 1, 0, 2, 5, 1, 0, 1, 0, 1, 0, 2, 2, 1, 0, 0, 1, 0, 0, 2, 4, 1, 0, 2, 4, 1, 0, 2, 3, 1, 0, 2, 2, 1, 0, 0, 2, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 6, 1, 0, 2, 1, 0, 0, 2, 4, 1, 0, 2, 8, 1, 0, 2, 1, 0, 0, 2, 2, 1, 0, 0, 18, 1, 0, 2, 5, 1, 0, 1, 0, 1, 0, 2, 5, 1, 0, 1, 0, 0, 0

Offset

0,1

Comments

a(n) = 0 if n + 1 is not squarefree.

a(n) <= 2 if n is even.

a(n) <= 6 if n is not divisible by 3.

a(n) <= 20 if n is not divisible by 5.

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Example

a(4) = 2 because 2^2 + 4 = 8 = 2^3 is not squarefree, while 2^0 + 4 = 5 and 2^1 + 4 = 6 are squarefree.

Maple

f:= proc(k) local j;
  for j from 0 to 300 do if not numtheory:-issqrfree(2^j+k) then return j fi
  od;
FAIL
end proc:
map(f, [$0..100]);

Prog

(PARI) a(n) = my(k=0); while (issquarefree(2^k+n), k++); k; \\ Michel Marcus, Aug 27 2023
(Python)
from itertools import count
from sympy import factorint
def A365196(n): return next(k for k in count(0) if max(factorint((1<<k)+n).values(), default=0)>1) # Chai Wah Wu, Aug 28 2023

Crossrefs

Cf. A005117.

Sequence in context: A065862 A368495 A189117 * A372257 A253580 A020921

Adjacent sequences: A365193 A365194 A365195 * A365197 A365198 A365199

Keyword

nonn

Author

Robert Israel, Aug 25 2023

Status

approved