cp's OEIS Frontend

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

%I A348364 #43 Dec 15 2021 01:48:10
%S A348364 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,
%T A348364 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,
%U A348364 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
%N A348364 Number of vertices on the axis of symmetry of the symmetric representation of sigma(n).
%C A348364 The values can be 1 or 2.
%C A348364 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).
%C A348364 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).
%H A348364 Antti Karttunen, <a href="/A348364/b348364.txt">Table of n, a(n) for n = 1..16384</a>
%F A348364 a(n) = 1 + A347950(n).
%F A348364 a(n) = 2 - A348327(n).
%e A348364 For n = 2, 6 and 10 the symmetric representation of sigma(n) respectively looks like this:
%e A348364 .
%e A348364 .           _       _       _
%e A348364 .         _| |     | |     | |
%e A348364 .    2   |_ _|     | |     | |
%e A348364 .               _ _| |     | |
%e A348364 .              |    _|     | |
%e A348364 .         _ _ _|  _|    _ _| |
%e A348364 .    6   |_ _ _ _|     |  _ _|
%e A348364 .                   _ _|_|
%e A348364 .                  |  _|
%e A348364 .         _ _ _ _ _| |
%e A348364 .   10   |_ _ _ _ _ _|
%e A348364 .
%e A348364 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.
%e A348364 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.
%e A348364 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.
%t A348364 a[n_] := 1 + Boole[DivisorSum[n, 1 &, n/2 <= #^2 < 2*n &] > 0]; Array[a, 100] (* _Amiram Eldar_, Oct 17 2021 *)
%o A348364 (PARI)
%o A348364 A347950(n) = ((sumdiv(n, d, my(d2 = d^2); (n/2 < d2) && (d2 <= n<<1))) > 0); \\ From A347950
%o A348364 A348364(n) = (1+A347950(n)); \\ _Antti Karttunen_, Dec 13 2021
%Y A348364 Parity gives A348327.
%Y A348364 Companion of A348406.
%Y A348364 Cf. A067742, A071090, A071540, A071562, A071563, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A240542, A281007, A299761, A303297, A340833, A346868, A347950.
%K A348364 nonn
%O A348364 1,1
%A A348364 _Omar E. Pol_, Oct 15 2021