A377321 a(n) is the smallest positive integer k such that n + k and n - k have the same number of distinct prime factors.

1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 3, 2, 2, 3, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 4, 4, 1, 1, 2, 1, 2, 1, 3, 3, 1, 3, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 3, 1, 4, 6, 3, 1, 3, 3, 2, 2, 2, 2, 3, 1, 1, 2, 1, 1, 3, 2, 3, 1, 1, 1, 3, 2, 3, 1, 1, 3, 2, 2

Offset: 4

Author

Felix Huber, Nov 17 2024

Keywords

• nonn

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.

Examples

			a(7) = 2 because 9 and 5 have both one distinct prime factor. 8 and 6 have a different number of distinct prime factors.
		

Links

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

Crossrefs

Cf. A001221, A082467, A188348, A377319, A377320.

Programs

Maple

A377321:=proc(n)
   local k;
   for k to n-1 do
      if NumberTheory:-Omega(n+k,'distinct')=NumberTheory:-Omega(n-k,'distinct') then
         return k
      fi
   od;
end proc;
seq(A377321(n),n=4..90);

Mathematica

A377321[n_] := Module[{k = 0}, While[PrimeNu[++k + n] != PrimeNu[n - k]]; k];
Array[A377321, 100, 4] (* Paolo Xausa, Dec 02 2024 *)

PARI

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

Formula

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