A377320 - OEIS

%I #15 Dec 02 2024 11:03:35

%S 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,

%T 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,

%U 1,3,4,4,3,1,4,13,6,2,5,15,2,7,1,3,3,1,3

%N a(n) is the smallest positive integer k such that n + k and n - k have the same number of 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="/A377320/b377320.txt">Table of n, a(n) for n = 4..10000</a>

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

%e 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.

%p A377320:=proc(n)

%p    local k;

%p    for k to n-1 do

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

%p          return k

%p       fi

%p    od;

%p end proc;

%p seq(A377320(n),n=4..90);

%t A377320[n_] := Module[{k = 0}, While[PrimeOmega[++k + n] != PrimeOmega[n - k]]; k];

%t Array[A377320, 100, 4] (* _Paolo Xausa_, Dec 02 2024 *)

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

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

%K nonn,new

%O 4,4

%A _Felix Huber_, Nov 17 2024