A348364 Number of vertices on the axis of symmetry of the symmetric representation of sigma(n).

2, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1

Offset

1,1

Comments

The values can be 1 or 2.

If a(n) = 1 then the symmetric representation of sigma(n) has an even number of parts and n is a number that have no middle divisors (cf. A071561).

If a(n) = 2 then the symmetric representation of sigma(n) has an odd number of parts and n is a number that have middle divisors (cf. A071562). The distance between both vertices divided by sqrt(2) equals the number of middle divisors of n (cf. A067742).

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

a(n) = 1 + A347950(n).

a(n) = 2 - A348327(n).

Example

For n = 2, 6 and 10 the symmetric representation of sigma(n) respectively looks like this:
.
.           _       _       _
.         _| |     | |     | |
.    2   |_ _|     | |     | |
.               _ _| |     | |
.              |    _|     | |
.         _ _ _|  _|    _ _| |
.    6   |_ _ _ _|     |  _ _|
.                   _ _|_|
.                  |  _|
.         _ _ _ _ _| |
.   10   |_ _ _ _ _ _|
.
For n = 2 there are two vertices on the axis of symmetry hence the symmetric representation of sigma(2) has an odd number of parts and 2 is a number that have middle divisors. The distance between both vertices divided by sqrt(2) equals the number of middle divisors of 2, that is A067742(2) = 1.
For n = 6 there are two vertices on the axis of symmetry so the symmetric representation of sigma(6) has an odd number of parts and 6 is a number that have middle divisors. The distance between both vertices divided by sqrt(2) equals the number of middle divisors of 6, that is A067742(6) = 2.
For n = 10 there is only one vertex on the axis of symmetry hence the symmetric representation of sigma(10) has an even number of parts and 10 is a number that have middle no divisors, so A067742(10) = 0.

Mathematica

a[n_] := 1 + Boole[DivisorSum[n, 1 &, n/2 <= #^2 < 2*n &] > 0]; Array[a, 100] (* Amiram Eldar, Oct 17 2021 *)

Prog

(PARI)
A347950(n) = ((sumdiv(n, d, my(d2 = d^2); (n/2 < d2) && (d2 <= n<<1))) > 0); \\ From A347950
A348364(n) = (1+A347950(n)); \\ Antti Karttunen, Dec 13 2021

Crossrefs

Parity gives A348327.

Companion of A348406.

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

Sequence in context: A346707 A138702 A344339 * A279620 A278109 A216665

Adjacent sequences: A348361 A348362 A348363 * A348365 A348366 A348367

Keyword

nonn

Author

Omar E. Pol, Oct 15 2021

Status

approved