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

1,2

Comments

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

Links

Table of n, a(n) for n=1..16.

Example

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].

Mathematica

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 *)

Crossrefs

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

Sequence in context: A062811 A363693 A236856 * A318843 A001470 A118932

Adjacent sequences: A338531 A338532 A338533 * A338535 A338536 A338537

Keyword

nonn,more

Author

Hartmut F. W. Hoft, Nov 01 2020

Status

approved