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

Author

Omar E. Pol, Oct 15 2021

Keywords

• nonn

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).

Examples

			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.
		

Links

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

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.

Programs

Mathematica

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

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

Formula

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

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