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A320521 a(n) is the smallest even number k such that the symmetric representation of sigma(k) has n parts.

%I A320521 #35 Dec 17 2024 08:39:34
%S A320521 2,10,50,230,1150,5050,22310,106030,510050,2065450,10236350
%N A320521 a(n) is the smallest even number k such that the symmetric representation of sigma(k) has n parts.
%C A320521 It appears that a(n) = 2 * q where q is odd and that the symmetric representation of sigma(a(n)/2) has the same number of parts as that for a(n). Number a(12) > 15000000. - _Hartmut F. W. Hoft_, Sep 22 2021
%e A320521 a(1) = 2 because the second row of A237593 is [2, 2], and the first row of the same triangle is [1, 1], therefore between both symmetric Dyck paths there is only one part: [3], equaling the sum of the divisors of 2: 1 + 2 = 3. See below:
%e A320521 .
%e A320521 .     _ _ 3
%e A320521 .    |_  |
%e A320521 .      |_|
%e A320521 .
%e A320521 .
%e A320521 a(2) = 10 because the 10th row of A237593 is [6, 2, 1, 1, 1, 1, 2, 6], and the 9th row of the same triangle is [5, 2, 2, 2, 2, 5], therefore between both symmetric Dyck paths there are two parts: [9, 9]. Also there are no even numbers k < 10 whose symmetric representation of sigma(k) has two parts. Note that the sum of these parts is 9 + 9 = 18, equaling the sum of the divisors of 10: 1 + 2 + 5 + 10 = 18. See below:
%e A320521 .
%e A320521 .     _ _ _ _ _ _ 9
%e A320521 .    |_ _ _ _ _  |
%e A320521 .              | |_
%e A320521 .              |_ _|_
%e A320521 .                  | |_ _ 9
%e A320521 .                  |_ _  |
%e A320521 .                      | |
%e A320521 .                      | |
%e A320521 .                      | |
%e A320521 .                      | |
%e A320521 .                      |_|
%e A320521 .
%e A320521 a(3) = 50 because the 50th row of A237593 is [26, 9, 4, 3, 3, 1, 2, 1, 1, 1, 1, 2, 1, 3, 3, 4, 9, 26], and the 49th row of the same triangle is [25, 9, 5, 3, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 3, 5, 9, 25], therefore between both symmetric Dyck paths there are three parts: [39, 15, 39]. Also there are no even numbers k < 50 whose symmetric representation of sigma(k) has three parts. Note that the sum of these parts is 39 + 15 + 39 = 93, equaling the sum of the divisors of 50: 1 + 2 + 5 + 10 + 25 + 50 = 93. (The diagram of the symmetric representation of sigma(50) = 93 is too large to include.)
%t A320521 (* support functions are defined in A341969, A341970 & A341971 *)
%t A320521 a320521[n_, len_] := Module[{list=Table[0, len], i, v}, For[i=2, i<=n, i+=2, v=Count[a341969[i], 0]+1;If[list[[v]]==0, list[[v]]=i]]; list]
%t A320521 a320521[15000000,11] (* _Hartmut F. W. Hoft_, Sep 22 2021 *)
%Y A320521 Row 1 of A320537.
%Y A320521 Cf. A237270 (the parts), A237271 (number of parts), A174973 (one part), A239929 (two parts), A279102 (three parts), A280107 (four parts), A320066 (five parts), A320511 (six parts).
%Y A320521 Cf. A000203, A018262, A005843, A196020, A235791, A236104, A237048, A237591, A237593, A239663, A239665, A240062, A245092, A262626, A296508.
%Y A320521 Cf. A250070, A262045, A341969, A341970, A341971, A347979.
%K A320521 nonn,more,hard
%O A320521 1,1
%A A320521 _Omar E. Pol_, Oct 14 2018
%E A320521 a(6)-a(11) from _Hartmut F. W. Hoft_, Sep 22 2021