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A377319 a(n) is the smallest positive integer k such that n + k and n - k have the same number of divisors.

%I A377319 #14 Dec 03 2024 23:33:13
%S A377319 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 A377319 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 A377319 1,4,4,4,3,6,4,9,6,2,5,5,2,8,1,3,3,2,3
%N A377319 a(n) is the smallest positive integer k such that n + k and n - k have the same number of divisors.
%C A377319 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 A377319 Felix Huber, <a href="/A377319/b377319.txt">Table of n, a(n) for n = 4..10000</a>
%F A377319 1 <= a(n) <= A082467(n).
%e A377319 a(8) = 2 because 10 and 6 have both four divisors. 9 and 7 have a different number of divisors.
%p A377319 A377319:=proc(n)
%p A377319    local k;
%p A377319    for k to n-1 do
%p A377319       if NumberTheory:-tau(n+k)=NumberTheory:-tau(n-k) then
%p A377319          return k
%p A377319       fi
%p A377319    od;
%p A377319 end proc;
%p A377319 seq(A377319(n),n=4..90);
%t A377319 A377319[n_] := Module[{k = 0}, While[DivisorSigma[0, ++k + n] != DivisorSigma[0, n - k]]; k];
%t A377319 Array[A377319, 100, 4] (* _Paolo Xausa_, Dec 03 2024 *)
%o A377319 (PARI) a(n) = my(k=1); while (numdiv(n+k) != numdiv(n-k), k++); k; \\ _Michel Marcus_, Nov 17 2024
%Y A377319 Cf. A000005, A067888 , A082467, A171937, A377320, A377321.
%K A377319 nonn
%O A377319 4,2
%A A377319 _Felix Huber_, Nov 17 2024