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

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

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

Maple

A377319:=proc(n)
   local k;
   for k to n-1 do
      if NumberTheory:-tau(n+k)=NumberTheory:-tau(n-k) then
         return k
      fi
   od;
end proc;
seq(A377319(n), n=4..90);

Mathematica

A377319[n_] := Module[{k = 0}, While[DivisorSigma[0, ++k + n] != DivisorSigma[0, n - k]]; k];
Array[A377319, 100, 4] (* Paolo Xausa, Dec 03 2024 *)

Prog

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

Crossrefs

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

Sequence in context: A364784 A239000 A105248 * A289495 A076302 A104524

Adjacent sequences: A377316 A377317 A377318 * A377320 A377321 A377322

Keyword

nonn

Author

Felix Huber, Nov 17 2024

Status

approved