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

4,2

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A189024 A304720 A057557 * A353364 A328390 A325599

Adjacent sequences: A377318 A377319 A377320 * A377323 A377324 A377325

Keyword

nonn,new

Author

Felix Huber, Nov 17 2024

Status

approved