login
A348364
Number of vertices on the axis of symmetry of the symmetric representation of sigma(n).
4
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
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
KEYWORD
nonn
AUTHOR
Omar E. Pol, Oct 15 2021
STATUS
approved