A377320 a(n) is the smallest positive integer k such that n + k and n - k have the same number of prime factors.

1, 1, 1, 3, 2, 2, 2, 6, 1, 5, 3, 2, 3, 6, 1, 1, 3, 2, 9, 2, 2, 5, 3, 4, 6, 1, 1, 11, 6, 4, 1, 6, 2, 2, 2, 2, 3, 8, 1, 1, 3, 2, 4, 3, 4, 12, 1, 1, 3, 2, 3, 1, 1, 3, 2, 7, 1, 4, 7, 4, 3, 6, 5, 1, 2, 1, 3, 5, 1, 3, 4, 4, 3, 1, 4, 13, 6, 2, 5, 15, 2, 7, 1, 3, 3, 1, 3

Offset

4,4

Comments

If the strong Goldbach conjecture is true, that every even number >= 8 is the sum of two distinct primes, then a positive integer k <= A082467(n) exists for n >= 4.

Links

Felix Huber, Table of n, a(n) for n = 4..10000

Formula

1 <= a(n) <= A082467(n).

Example

a(7) = 3 because 10 and 4 have both two prime factors. 8 and 6 or 9 and 7 respectively have a different number of prime factors.

Maple

A377320:=proc(n)
   local k;
   for k to n-1 do
      if NumberTheory:-Omega(n+k)=NumberTheory:-Omega(n-k) then
         return k
      fi
   od;
end proc;
seq(A377320(n), n=4..90);

Mathematica

A377320[n_] := Module[{k = 0}, While[PrimeOmega[++k + n] != PrimeOmega[n - k]]; k];
Array[A377320, 100, 4] (* Paolo Xausa, Dec 02 2024 *)

Prog

(PARI) a(n) = my(k=1); while (bigomega(n+k) != bigomega(n-k), k++); k; \\ Michel Marcus, Nov 17 2024

Crossrefs

Cf. A001222, A082467, A178139, A377319, A377321.

Sequence in context: A175333 A130845 A134653 * A090207 A364571 A202538

Adjacent sequences: A377317 A377318 A377319 * A377321 A377323 A377324

Keyword

nonn,new

Author

Felix Huber, Nov 17 2024

Status

approved