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

1,4

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.

Links

Table of n, a(n) for n=1..105.

Formula

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

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 *)

Crossrefs

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

Sequence in context: A306737 A178474 A164822 * A110627 A205107 A228667

Adjacent sequences: A361205 A361206 A361207 * A361209 A361210 A361211

Keyword

nonn

Author

Omar E. Pol, Mar 06 2023

Status

approved