This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A340833 #66 Nov 01 2021 01:08:33
%S A340833 4,6,7,10,9,12,11,14,14,15,13,18,13,17,20,22,15,22,15,22,23,21,17,26,
%T A340833 22,21,25,28,19,30,19,30,27,23,26,32,21,25,29,34,21,34,21,33,36,27,23,
%U A340833 38,30,38,31,35,23,38,35,42,33,29,25,42,25,29,42,42,37,44,27
%N A340833 a(n) is the number of vertices in the diagram of the symmetric representation of sigma(n).
%C A340833 If A237271(n) is odd then a(n) is even.
%C A340833 If A237271(n) is even then a(n) is odd.
%C A340833 The above sentences arise that the diagram is always symmetric for any value of n hence the number of edges is always an even number. Also from Euler's formula.
%C A340833 Indices of odd terms give A071561.
%C A340833 Indices of even terms give A071562.
%C A340833 For another version with subparts see A340847 from which first differs at a(6).
%C A340833 The parity of this sequence is also the characteristic function of numbers that have no middle divisors (cf. A348327). - _Omar E. Pol_, Oct 14 2021
%H A340833 Michael De Vlieger, <a href="/A340833/b340833.txt">Table of n, a(n) for n = 1..10000</a>
%H A340833 Michael De Vlieger, <a href="/A340833/a340833.png">Log-log scatterplot of a(n)</a> for n=1..10^4, accentuating a(m) for m=1..2^8 for clarity, and labeling a(k) for k=1..24.
%F A340833 a(n) = A340846(n) - A237271(n) + 1 (Euler's formula).
%e A340833 Illustration of initial terms:
%e A340833 . _ _ _ _
%e A340833 . _ _ _ |_ _ _ |_
%e A340833 . _ _ _ |_ _ _| | |_
%e A340833 . _ _ |_ _ |_ |_ _ |_ _ |
%e A340833 . _ _ |_ _|_ |_ | | | | |
%e A340833 . _ |_ | | | | | | | | |
%e A340833 . |_| |_| |_| |_| |_| |_|
%e A340833 .
%e A340833 n: 1 2 3 4 5 6
%e A340833 a(n): 4 6 7 10 9 12
%e A340833 .
%e A340833 For n = 6 the diagram has 12 vertices so a(6) = 12.
%e A340833 On the other hand the diagram has 12 edges and only one part or region, so applying Euler's formula we have that a(6) = 12 - 1 + 1 = 12.
%e A340833 . _ _ _ _ _
%e A340833 . _ _ _ _ _ |_ _ _ _ _|
%e A340833 . _ _ _ _ |_ _ _ _ | |_ _
%e A340833 . |_ _ _ _| | |_ |_ |
%e A340833 . |_ |_ |_ _ |_|_ _
%e A340833 . |_ _ |_ _ | | |
%e A340833 . | | | | | |
%e A340833 . | | | | | |
%e A340833 . | | | | | |
%e A340833 . |_| |_| |_|
%e A340833 .
%e A340833 n: 7 8 9
%e A340833 a(n): 11 14 14
%e A340833 .
%e A340833 For n = 9 the diagram has 14 vertices so a(9) = 14.
%e A340833 On the other hand the diagram has 16 edges and three parts or regions, so applying Euler's formula we have that a(9) = 16 - 3 + 1 = 14.
%e A340833 Another way for the illustration of initial terms is as follows:
%e A340833 --------------------------------------------------------------------------
%e A340833 . n a(n) Diagram
%e A340833 --------------------------------------------------------------------------
%e A340833 _
%e A340833 1 4 |_| _
%e A340833 _| | _
%e A340833 2 6 |_ _| | | _
%e A340833 _ _|_| | | _
%e A340833 3 7 |_ _| _| | | | _
%e A340833 _ _| _| | | | | _
%e A340833 4 10 |_ _ _| _|_| | | | | _
%e A340833 _ _ _| _ _| | | | | | _
%e A340833 5 9 |_ _ _| | _| | | | | | | _
%e A340833 _ _ _| _| _|_| | | | | | | _
%e A340833 6 12 |_ _ _ _| _| _ _| | | | | | | | _
%e A340833 _ _ _ _| _| _ _| | | | | | | | | _
%e A340833 7 11 |_ _ _ _| | _| _ _|_| | | | | | | | | _
%e A340833 _ _ _ _| | _| | _ _| | | | | | | | | | _
%e A340833 8 14 |_ _ _ _ _| |_ _| | _ _| | | | | | | | | | | _
%e A340833 _ _ _ _ _| _ _|_| _ _|_| | | | | | | | | | |
%e A340833 9 14 |_ _ _ _ _| | _| _| _ _ _| | | | | | | | | |
%e A340833 _ _ _ _ _| | _| _| _ _| | | | | | | | |
%e A340833 10 15 |_ _ _ _ _ _| | _| | _ _|_| | | | | | |
%e A340833 _ _ _ _ _ _| | _| | _ _ _| | | | | |
%e A340833 11 13 |_ _ _ _ _ _| | _ _| _| | _ _ _| | | | |
%e A340833 _ _ _ _ _ _| | _ _| _|_| _ _ _|_| | |
%e A340833 12 18 |_ _ _ _ _ _ _| | _ _| _ _| | _ _ _| |
%e A340833 _ _ _ _ _ _ _| | _| | _| | _ _ _|
%e A340833 13 13 |_ _ _ _ _ _ _| | | _| _| _| |
%e A340833 _ _ _ _ _ _ _| | |_ _| _| _|
%e A340833 14 17 |_ _ _ _ _ _ _ _| | _ _| _|
%e A340833 _ _ _ _ _ _ _ _| | _ _|
%e A340833 15 20 |_ _ _ _ _ _ _ _| | |
%e A340833 _ _ _ _ _ _ _ _| |
%e A340833 16 22 |_ _ _ _ _ _ _ _ _|
%e A340833 ...
%t A340833 MapAt[# + 1 &, #, 1] &@ Map[Length@ Union[Join @@ #] - 1 &, Partition[Prepend[#, {{0, 0}}], 2, 1]] &@ Table[{{0, 0}}~Join~Accumulate[Join[#, Reverse[Reverse /@ (-1*#)]]] &@ MapIndexed[Which[#2 == 1, {#1, 0}, Mod[#2, 2] == 0, {0, #1}, True, {-#1, 0}] & @@ {#1, First[#2]} &, If[Length[#] == 0, {n, n}, Join[{n}, #, {n - Total[#]}]]] &@ Differences[n - Array[(Ceiling[(n + 1)/# - (# + 1)/2]) &, Floor[(Sqrt[8 n + 1] - 1)/2]]], {n, 67}] (* _Michael De Vlieger_, Oct 27 2021 *)
%Y A340833 Parity gives A348327.
%Y A340833 Cf. A237271 (number of parts or regions).
%Y A340833 Cf. A340846 (number of edges).
%Y A340833 Cf. A340847 (number of vertices in the diagram with subparts).
%Y A340833 Cf. A294723 (total number of vertices in the unified diagram).
%Y A340833 Cf. A239931-A239934 (illustration of first 32 diagrams).
%Y A340833 Cf. A000203, A071561, A071562, A196020, A236104, A235791, A237048, A237270, A237590, A237591, A237593, A239660, A245092, A262626, A340848.
%K A340833 nonn,look
%O A340833 1,1
%A A340833 _Omar E. Pol_, Jan 23 2021
%E A340833 Terms a(33) and beyond from _Michael De Vlieger_, Oct 27 2021