A377319 - OEIS

%I #14 Dec 03 2024 23:33:13

%S 1,2,1,1,2,1,3,3,1,6,3,2,3,6,1,1,3,2,9,5,2,6,3,3,6,12,1,4,6,4,1,5,2,2,

%T 6,2,3,1,1,8,3,2,11,3,4,7,3,1,6,2,3,1,1,4,7,9,1,4,7,4,3,6,5,2,2,2,3,6,

%U 1,4,4,4,3,6,4,9,6,2,5,5,2,8,1,3,3,2,3

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

%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="/A377319/b377319.txt">Table of n, a(n) for n = 4..10000</a>

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

%e a(8) = 2 because 10 and 6 have both four divisors. 9 and 7 have a different number of divisors.

%p A377319:=proc(n)

%p    local k;

%p    for k to n-1 do

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

%p          return k

%p       fi

%p    od;

%p end proc;

%p seq(A377319(n),n=4..90);

%t A377319[n_] := Module[{k = 0}, While[DivisorSigma[0, ++k + n] != DivisorSigma[0, n - k]]; k];

%t Array[A377319, 100, 4] (* _Paolo Xausa_, Dec 03 2024 *)

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

%Y Cf. A000005, A067888 , A082467, A171937, A377320, A377321.

%K nonn,new

%O 4,2

%A _Felix Huber_, Nov 17 2024