A348406 Number of vertices on the axis of symmetry of the symmetric representation of sigma(n) with subparts.

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

Offset

1,1

Comments

The number of middle divisors of n is equal to a(n) - 1.

For the definition of "subparts" see A279387.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(n) = 1 + A067742(n).

Example

For n = 2, 6 and 10 the symmetric representation of sigma(n) with subparts respectively looks like this:
.
.           _       _       _
.         _| |     | |     | |
.    2   |_ _|     | |     | |
.               _ _| |     | |
.              |  _ _|     | |
.         _ _ _| |_|    _ _| |
.    6   |_ _ _ _|     |  _ _|
.                   _ _|_|
.                  |  _|
.         _ _ _ _ _| |
.   10   |_ _ _ _ _ _|
.
For n = 2 there are two vertices on the axis of symmetry hence the number of middle divisors of 2 is equal to 2 - 1 = 1.
For n = 6 there are three vertices on the axis of symmetry hence the number of middle divisors of 6 is equal to 3 - 1 = 2.
For n = 10 there is only one vertex on the axis of symmetry hence the number of middle divisors of 10 is equal to 1 - 1 = 0.

Mathematica

a[n_] := 1 + DivisorSum[n, 1 &, n/2 <= #^2 < 2*n &]; Array[a, 100] (* Amiram Eldar, Oct 17 2021 *)

Prog

(PARI)
A067742(n) = sumdiv(n, d, my(d2 = d^2); n / 2 < d2 && d2 <= n << 1); \\ From A067742
A348406(n) = (1 + A067742(n));

Crossrefs

Companion of A348364.

Cf. A067742, A071090, A071540, A071562, A071563, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A240542, A279387, A281007, A299761, A303297, A340833, A340847, A346868, A347950, A348327.

Sequence in context: A105153 A000924 A306911 * A344165 A305432 A305298

Adjacent sequences: A348403 A348404 A348405 * A348407 A348408 A348409

Keyword

nonn

Author

Omar E. Pol, Oct 17 2021

Status

approved