A361208 Number of middle divisors of the n-th number whose divisors increase by a factor of 2 or less.

1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 3, 2, 2, 2, 2, 2, 1, 2, 2, 2, 4, 2, 1, 2, 2, 3, 2, 2, 2, 1, 2, 2, 4, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 1, 2, 4, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 1, 2, 3, 2, 2, 2, 4, 4, 2, 2, 3, 2, 2, 2, 2, 2, 2, 4

Offset: 1

Author

Omar E. Pol, Mar 06 2023

Keywords

• nonn

Comments

The middle divisors of n are the divisors in the half-open interval [sqrt(n/2), sqrt(n*2)).

Also consider the n-th number k with the property that the symmetric representation of sigma(k) has only one part. a(n) is the number of square cells on the axis of symmetry of the diagram.

For the diagrams related to the first 13 terms of this sequence see A317305.

Crossrefs

Cf. A067742, A174973, A237048, A237270, A237593, A281007, A317305, A354452.

Programs

Mathematica

With[{$MaxExtraPrecision = 1000}, Map[Function[n, Count[Divisors[n], ?(Sqrt[n/2] <= # < Sqrt[2 n] &)]], Select[Range[500], And @@ Map[# <= 2 &, (Rest[#]/Most[#])] &@ Divisors[#] &]] ]  (* _Michael De Vlieger, Mar 28 2023, after Harvey P. Dale at A174973 *)

Formula

a(n) = A067742(A174973(n)).