A239663 -id:A239663 - cp's OEIS frontend

A338534 a(n) is the smallest number k whose symmetric representation of sigma(k) consists of n regions (or parts) and whose areas are strictly decreasing towards the diagonal.

1, 3, 9, 21, 81, 147, 441, 903, 2499, 6069, 15351, 29095, 80367, 153065, 366025, 643885

Offset: 1

Hartmut F. W. Hoft, Nov 01 2020

For a(n) <= 750000 the regions in the symmetric representation of sigma(a(n)) have at most width of 2.

			a(5) = 81 is in this sequence since its symmetric representation of sigma is the first to consist of 5 regions [41, 15, 9, 15, 41] decreasing towards the diagonal.
63 is the only number smaller than 81 with 5 regions, but is not in the sequence since the regions of its symmetric representation of sigma are [32, 12, 16, 12, 32].

Cf. A237048, A237270, A237271, A237593, A239663, A318843.

cd[n_, k_] := If[Divisible[n, k], 1, 0]
legs[n_, len_] := Module[{tL = Map[Ceiling[(n + 1)/# - (# + 1)/2] &, Range[len]]}, tL - Append[Rest[tL], 0]]
a237048[n_, len_] := Map[If[OddQ[#], cd[n, #], cd[n-#/2, #]]&, Range[len]]
widths[n_, len_] := Drop[FoldList[Plus, 0, Map[(-1)^(#+1)&, Range[len]] a237048[n, len]], 1]
regions[n_] := Module[{r=Floor[(Sqrt[8n+1]-1)/2], wL, wM, diag, sL, sLs, regs, lens}, wL=widths[n, r]; wM=Max[wL]; diag=Last[wL]; sL=legs[n, r]wL; sLs=SplitBy[sL, #!=0&]; regs=Select[Map[Fold[Plus, 0, #]&, sLs], #!=0&]; lens=Length[regs]; If[diag==0, {wM, Join[regs, Reverse[regs]]}, {wM, Join[Most[regs], {2Last[regs]-diag}, Reverse[Most[regs]]]}]]
a338534[n_, b_] := Module[{k, r, len, s, list=Table[0, b]}, For[k=1, k<=n, k++, r=Last[regions[k]]; len=Length[r]; s=Take[r, Floor[(len+1)/2]]; If[AllTrue[Most[s]-Rest[s], #>0&]&&list[[len]]==0, list[[len]]=k]]; list]
Take[a338534[750000, 20], 16] (* sequence data *)

A357775 Numbers k with the property that the symmetric representation of sigma(k) has seven parts.

Original entry on oeis.org

357, 399, 441, 483, 513, 567, 609, 621, 651, 729, 759, 777, 783, 837, 861, 891, 957, 999, 1023, 1053, 1089, 1107, 1131, 1161, 1209, 1221, 1269, 1287, 1323, 1353, 1419, 1431, 1443, 1521, 1551, 1595, 1599, 1677, 1705, 1749, 1815, 1833, 1887, 1947, 1989, 2013, 2035, 2067, 2091, 2145, 2193, 2223, 2255

Offset: 1

Omar E. Pol, Oct 12 2022

nonn

			357 is in the sequence because the 357th row of A237593 is [179, 60, 31, 18, 12, 9, 7, 6, 4, 4, 3, 3, 2, 3, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 3, 2, 3, 3, 4, 4, 6, 7, 9, 12, 18, 31, 60, 179], and the 356th row of the same triangle is [179, 60, 30, 18, 13, 9, 6, 6, 4, 4, 3, 3, 3, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 3, 3, 3, 4, 4, 6, 6, 9, 13, 18, 30, 60, 179], therefore between both symmetric Dyck paths there are seven parts: [179, 61, 29, 38, 29, 61, 179].
Note that the sum of these parts is 179 + 61 + 29 + 38 + 29 + 61 + 179 = 576, equaling the sum of the divisors of 357: 1 + 3 + 7 + 17 + 21 + 51 + 119 + 357 = 576.
(The diagram of the symmetric representation of sigma(357) = 576 is too large to include.)

Column 7 of A240062.
Cf. A237270 (the parts), A237271 (number of parts), A238443 = A174973 (one part), A239929 (two parts), A279102 (three parts), A280107 (four parts), A320066 (five parts), A320511 (six parts).
Cf. A018411, A196020, A236104, A235791, A237591, A237593, A239663, A266094.

A237271(a(n)) = 7.

cp's OEIS Frontend

A338534 a(n) is the smallest number k whose symmetric representation of sigma(k) consists of n regions (or parts) and whose areas are strictly decreasing towards the diagonal.

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