A389426 a(n) is the least k such that k*(n-k) + 1 is prime, or -1 if there is no such k.

-1, 1, 1, 2, 1, -1, 1, 2, 3, 2, 1, 6, 1, 4, 3, 2, 1, 6, 1, 2, 9, 2, 1, 6, 2, 4, 3, 2, 1, -1, 1, 2, 6, 10, 2, 6, 1, 2, 3, 8, 1, 18, 1, 6, 3, 2, 1, 12, 3, 2, 6, 2, 1, -1, 2, 2, 3, 2, 1, 12, 1, 4, 3, 4, 2, 24, 1, 4, 3, 2, 1, 6, 1, 4, 12, 2, 2, 6, 1, 2, 12, 4, 1, 30, 2, 10, 6, 2, 1, 12, 2, 2, 3, 16

Offset

1,4

Comments

a(n) = -1 for n in A383636, 1 for n prime, 2 for composites in A098090.

If n is even then a(n) is -1 or even.

If n is divisible by 3 then a(n) is -1 or divisible by 3.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

a(4) = 2 because 2*(4-2) + 1 = 5 is prime but 1*(4-1) + 1 = 4 is not.

Maple

f:= proc(n) local k;
  for k from 1 to n/2 do
    if isprime(k*(n-k)+1) then return k fi
  od;
  -1
end proc:
map(f, [$1..100]);

Prog

(Python)
from sympy import isprime
def A389426(n): return next((k for k in range(1, (n>>1)+1) if isprime(k*(n-k)+1)), -1) # Chai Wah Wu, Oct 06 2025

Crossrefs

Cf. A098090, A109905, A383636.

Sequence in context: A039958 A029344 A375731 * A316854 A367759 A125769

Adjacent sequences: A389423 A389424 A389425 * A389427 A389428 A389429

Keyword

sign

Author

Robert Israel, Oct 03 2025

Status

approved