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%I A340847 #37 May 10 2025 16:19:15 %S A340847 4,6,7,10,9,13,11,14,14,15,13,23,13,17,21,22,15,26,15,25,23,21,27,35, %T A340847 22,21,25,29,19,41,19,30 %N A340847 a(n) is the number of vertices in the diagram of the symmetric representation of sigma(n) with subparts. %C A340847 Theorem: Indices of even terms give A028982. Indices of odd terms give A028983. %C A340847 If A001227(n) is odd then a(n) is even. %C A340847 If A001227(n) is even then a(n) is odd. %C A340847 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 A340847 For another version see A340833 from which first differs at a(6). %C A340847 For the definition of subparts see A279387. For more information about the subparts see also A237271, A280850, A280851, A296508, A335616. %C A340847 Note that in this version of the diagram of the symmetric representation of sigma(n) all regions are called "subparts". The number of subparts equals A001227(n). %F A340847 a(n) = A340848(n) - A001227(n) + 1 (Euler's formula). %e A340847 Illustration of initial terms: %e A340847 . _ _ _ _ %e A340847 . _ _ _ |_ _ _ |_ %e A340847 . _ _ _ |_ _ _| | |_|_ %e A340847 . _ _ |_ _ |_ |_ _ |_ _ | %e A340847 . _ _ |_ _|_ |_ | | | | | %e A340847 . _ |_ | | | | | | | | | %e A340847 . |_| |_| |_| |_| |_| |_| %e A340847 . %e A340847 n: 1 2 3 4 5 6 %e A340847 a(n): 4 6 7 10 9 13 %e A340847 . %e A340847 For n = 6 the diagram has 13 vertices so a(6) = 13. %e A340847 On the other hand the diagram has 14 edges and two subparts or regions, so applying Euler's formula we have that a(6) = 14 - 2 + 1 = 13. %e A340847 . %e A340847 . _ _ _ _ _ %e A340847 . _ _ _ _ _ |_ _ _ _ _| %e A340847 . _ _ _ _ |_ _ _ _ | |_ _ %e A340847 . |_ _ _ _| | |_ |_ | %e A340847 . |_ |_ |_ _ |_|_ _ %e A340847 . |_ _ |_ _ | | | %e A340847 . | | | | | | %e A340847 . | | | | | | %e A340847 . | | | | | | %e A340847 . |_| |_| |_| %e A340847 . %e A340847 n: 7 8 9 %e A340847 a(n): 11 14 14 %e A340847 . %e A340847 For n = 9 the diagram has 14 vertices so a(9) = 14. %e A340847 On the other hand the diagram has 16 edges and three subparts or regions, so applying Euler's formula we have that a(9) = 16 - 3 + 1 = 14. %e A340847 Another way for the illustration of initial terms is as follows: %e A340847 -------------------------------------------------------------------------- %e A340847 . n a(n) Diagram %e A340847 -------------------------------------------------------------------------- %e A340847 _ %e A340847 1 4 |_| _ %e A340847 _| | _ %e A340847 2 6 |_ _| | | _ %e A340847 _ _|_| | | _ %e A340847 3 7 |_ _| _| | | | _ %e A340847 _ _| _| | | | | _ %e A340847 4 10 |_ _ _| _|_| | | | | _ %e A340847 _ _ _| _ _| | | | | | _ %e A340847 5 9 |_ _ _| | _ _| | | | | | | _ %e A340847 _ _ _| |_| _|_| | | | | | | _ %e A340847 6 13 |_ _ _ _| _| _ _| | | | | | | | _ %e A340847 _ _ _ _| _| _ _| | | | | | | | | _ %e A340847 7 11 |_ _ _ _| | _| _ _|_| | | | | | | | | _ %e A340847 _ _ _ _| | _| | _ _| | | | | | | | | | _ %e A340847 8 14 |_ _ _ _ _| |_ _| | _ _| | | | | | | | | | | _ %e A340847 _ _ _ _ _| _ _|_| _ _|_| | | | | | | | | | | %e A340847 9 14 |_ _ _ _ _| | _| _| _ _ _| | | | | | | | | | %e A340847 _ _ _ _ _| | _| _| _ _ _| | | | | | | | | %e A340847 10 15 |_ _ _ _ _ _| | _| _| | _ _|_| | | | | | | %e A340847 _ _ _ _ _ _| | _| _| | _ _ _| | | | | | %e A340847 11 13 |_ _ _ _ _ _| | |_ _| _| | _ _ _| | | | | %e A340847 _ _ _ _ _ _| | _ _| _|_| _ _ _|_| | | %e A340847 12 23 |_ _ _ _ _ _ _| | _ _| _ _| | _ _ _| | %e A340847 _ _ _ _ _ _ _| | _| | _ _| | _ _ _| %e A340847 13 13 |_ _ _ _ _ _ _| | | _| |_| _| | %e A340847 _ _ _ _ _ _ _| | |_ _| _| _| %e A340847 14 17 |_ _ _ _ _ _ _ _| | _ _| _| %e A340847 _ _ _ _ _ _ _ _| | _ _| %e A340847 15 21 |_ _ _ _ _ _ _ _| | | %e A340847 _ _ _ _ _ _ _ _| | %e A340847 16 22 |_ _ _ _ _ _ _ _ _| %e A340847 ... %Y A340847 Cf. A001227 (number of subparts or regions). %Y A340847 Cf. A340848 (number of edges). %Y A340847 Cf. A340833 (numer of vertices in the diagram only with parts). %Y A340847 Cf. A317293 (total number of vertices in the unified diagram). %Y A340847 Cf. A000203, A028982, A028983, A060831, A196020, A236104, A235791, A237048, A237270, A237591, A237593, A239660, A245092, A262626, A279387, A280850, A280851, A296508, A335616, A340846. %K A340847 nonn,more %O A340847 1,1 %A A340847 _Omar E. Pol_, Jan 24 2021