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A239663
a(n) is the smallest number k such that the symmetric representation of sigma(k) has n parts.
21
1, 3, 9, 21, 63, 147, 357, 903, 2499, 6069, 13915, 29095, 59455, 142945, 320045, 643885, 1367465, 3287735, 6779135, 13853015, 30262595, 61773745
OFFSET
1,2
COMMENTS
Conjecture: where records occur in A237271. - Omar E. Pol, Dec 27 2016
For more information about the symmetric representation of sigma see A237270, A237593.
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
It appears that all terms are odd numbers. - Omar E. Pol, Oct 14 2018
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
Observation: at least for n = 1..21 we have that 2*a(n) < a(n+1). - Omar E. Pol, Sep 22 2021
EXAMPLE
------------------------------------------------------
n a(n) A239665 A266094(n)
------------------------------------------------------
1 1 [1] 1
2 3 [2, 2] 4
3 9 [5, 3, 5] 13
4 21 [11, 5, 5, 11] 32
5 63 [32, 12, 16, 12, 32] 104
...
For n = 3 the symmetric representation of sigma(9) = 13 contains three parts [5, 3, 5] as shown below:
.
. _ _ _ _ _ 5
. |_ _ _ _ _|
. |_ _ 3
. |_ |
. |_|_ _ 5
. | |
. | |
. | |
. | |
. |_|
.
MATHEMATICA
(* a239663[] permits computation in intervals *)
(* Function a237270[] is defined in A237270 *)
(* variable "list" contains the first occurrences up to m *)
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]
a239663[{1}, {1, 1000}] (* computes the first 8 values *)
(* Hartmut F. W. Hoft, Jul 08 2014 *)
(* support functions are defined in A341969, A341970 & A341971 *)
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]
a239663[62000000, 22] (* Hartmut F. W. Hoft, Sep 22 2021 *)
KEYWORD
nonn,more,hard
AUTHOR
Omar E. Pol, Mar 23 2014
EXTENSIONS
a(6)-a(8) from Michel Marcus, Mar 28 2014
a(9) from Michel Marcus, Mar 29 2014
a(10)-a(11) from Michel Marcus, Apr 02 2014
a(12) from Hartmut F. W. Hoft, Jul 08 2014
a(13)-a(18) from Hartmut F. W. Hoft, Dec 09 2014
a(19)-a(22) from Hartmut F. W. Hoft, Sep 22 2021
STATUS
approved