A375391 a(n) is the greatest odd number k such that n^2 + j is a semiprime for all odd numbers j from 1 to k.

-1, -1, 1, -1, 1, -1, -1, 1, 1, -1, 1, 1, -1, -1, 1, -1, -1, -1, 1, -1, -1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, -1, -1, 3, 1, -1, -1, -1, 1, -1, -1, 1, -1, 9, 1, 3, -1, 3, 1, 1, 1, 1, -1, -1, -1, -1, -1, 1, 1, 3, 1, 1, -1, 1, 1, -1, -1, -1, 1, -1, 1, -1, -1, -1, -1, 1, -1, 3, 1, 3, -1, -1, -1, -1

Offset: 1

Author

Robert Israel, Aug 15 2024

Keywords

• sign

Comments

a(n) = -1 if n^2 + 1 is not a semiprime.

a(n) <= 1 if n is odd, since n^2 + 3 is divisible by 4.

a(n) <= 15 since one of n^2 + 1, n^2 + 3, ..., n^2 + 17 is divisible by 9.

First occurrences of values: a(3) = 1, a(34) = 3, a(152) = 5, a(102) = 7, a(44) = 9, a(824264) = 11, a(21394) = 13, a(121364) = 15.

Examples

			a(44) = 9 since 44^2 + 1 = 1937 = 13 * 149, 44^2 + 3 = 1939 = 7 * 277, 44^2 + 5 = 1941 = 3 * 647, 44^2 + 7 = 1943 = 29 * 67 and 44^2 + 9 = 1945 = 5 * 389 are all semiprimes but 44^2 + 11 = 1947 = 3 * 11 * 59 is not a semiprime.
		

Links

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

Crossrefs

Cf. A001358, A085722, A242331, A242333, A375390.

Programs

Maple

f:= proc(n) local i;
  for i from 1 by 2 while numtheory:-bigomega(n^2+i) = 2 do od:
  i-2
end proc:
map(f, [$1..100]);