A377321 - OEIS

%I #12 Dec 03 2024 12:26:46

%S 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,

%T 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,

%U 1,1,2,1,1,3,2,3,1,1,1,3,2,3,1,1,3,2,2

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

%C 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.

%H Felix Huber, <a href="/A377321/b377321.txt">Table of n, a(n) for n = 4..10000</a>

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

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

%p A377321:=proc(n)

%p    local k;

%p    for k to n-1 do

%p       if NumberTheory:-Omega(n+k,'distinct')=NumberTheory:-Omega(n-k,'distinct') then

%p          return k

%p       fi

%p    od;

%p end proc;

%p seq(A377321(n),n=4..90);

%t A377321[n_] := Module[{k = 0}, While[PrimeNu[++k + n] != PrimeNu[n - k]]; k];

%t Array[A377321, 100, 4] (* _Paolo Xausa_, Dec 02 2024 *)

%o (PARI) a(n) = my(k=1); while (omega(n+k) != omega(n-k), k++); k; \\ _Michel Marcus_, Nov 17 2024

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

%K nonn,new

%O 4,2

%A _Felix Huber_, Nov 17 2024