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%I #88 Sep 23 2021 12:01:30
%S 1,3,9,21,63,147,357,903,2499,6069,13915,29095,59455,142945,320045,
%T 643885,1367465,3287735,6779135,13853015,30262595,61773745
%N a(n) is the smallest number k such that the symmetric representation of sigma(k) has n parts.
%C Conjecture: where records occur in A237271. - _Omar E. Pol_, Dec 27 2016
%C For more information about the symmetric representation of sigma see A237270, A237593.
%C This sequence of (first occurrence of) parts appears to be strictly increasing in contrast to sequence A250070 of (first occurrence of) maximum widths. - _Hartmut F. W. Hoft_, Dec 09 2014
%C It appears that all terms are odd numbers. - _Omar E. Pol_, Oct 14 2018
%C Let n = 2^m * q with m>0 and q odd; then the 1's in even positions of row n in the triangle of A237048 are at positions 2^(m+1) * d <= row(n) where d divides q. For n/2 the even positions of 1's occur at the smaller values 2^m * d <= row(n/2), thus either keeping or reducing widths (A249223) of parts in the symmetric representation of sigma for n/2 inherited from row n. Therefore the number of parts for n is at most as large as for n/2, i.e., all numbers in this sequence are odd. - _Hartmut F. W. Hoft_, Sep 22 2021
%C Observation: at least for n = 1..21 we have that 2*a(n) < a(n+1). - _Omar E. Pol_, Sep 22 2021
%H Hartmut F. W. Hoft, <a href="/A239663/a239663.pdf">Procedural implementation for extension values</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Perspective view of the stepped pyramid (16 levels)</a>
%e ------------------------------------------------------
%e n a(n) A239665 A266094(n)
%e ------------------------------------------------------
%e 1 1 [1] 1
%e 2 3 [2, 2] 4
%e 3 9 [5, 3, 5] 13
%e 4 21 [11, 5, 5, 11] 32
%e 5 63 [32, 12, 16, 12, 32] 104
%e ...
%e For n = 3 the symmetric representation of sigma(9) = 13 contains three parts [5, 3, 5] as shown below:
%e .
%e . _ _ _ _ _ 5
%e . |_ _ _ _ _|
%e . |_ _ 3
%e . |_ |
%e . |_|_ _ 5
%e . | |
%e . | |
%e . | |
%e . | |
%e . |_|
%e .
%t (* a239663[] permits computation in intervals *)
%t (* Function a237270[] is defined in A237270 *)
%t (* variable "list" contains the first occurrences up to m *)
%t a239663[list_,{m_, n_}]:=Module[{firsts=list, g=Length[list], i, p}, For[i=m, i<=n, i++, p=Length[a237270[i]]; If[p>g, AppendTo[firsts, i]; g=p]]; firsts]
%t a239663[{1}, {1, 1000}] (* computes the first 8 values *)
%t (* _Hartmut F. W. Hoft_, Jul 08 2014 *)
%t (* support functions are defined in A341969, A341970 & A341971 *)
%t a239663[n_, len_] := Module[{list=Table[0, len], i, v}, For[i=1, i<=n, i+=2, v=Count[a341969[i], 0]+1;If[list[[v]]==0, list[[v]]=i]]; list]
%t a239663[62000000,22] (* _Hartmut F. W. Hoft_, Sep 22 2021 *)
%Y Row 1 of A240062.
%Y Cf. A000203, A196020, A236104, A235791, A237048, A237270, A237271, A237591, A237593, A238443, A239657, A239660, A239665, A239931-A239934, A245092, A262626, A266094.
%Y Cf. A249223, A250070, A262045, A320521, A341969, A341970, A341971, A347980.
%K nonn,more,hard
%O 1,2
%A _Omar E. Pol_, Mar 23 2014
%E a(6)-a(8) from _Michel Marcus_, Mar 28 2014
%E a(9) from _Michel Marcus_, Mar 29 2014
%E a(10)-a(11) from _Michel Marcus_, Apr 02 2014
%E a(12) from _Hartmut F. W. Hoft_, Jul 08 2014
%E a(13)-a(18) from _Hartmut F. W. Hoft_, Dec 09 2014
%E a(19)-a(22) from _Hartmut F. W. Hoft_, Sep 22 2021
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