A395533 a(n) is the first occurrence k of an uninterrupted sequence of n adjacent center parts of the symmetric representation of sigma of i, SRS(i), i = k to i = k + n - 1 where each SRS(i) has an odd number of parts and SRS(k-1) and SRS(k+n) each have an even number of parts.

4, 1, 48, 322, 1517, 3476, 45998, 405818, 591216, 43220379, 172187609

Offset

1,1

Comments

a(n) > 10^7 for all n > 9.

Conjecture: a(n) is increasing for n > 1.

By definition, for any a(n) = k adjacent center parts of SRS(i) and SRS(i+1), k <= i < k+n-1, share a common boundary of at least length 2 about the diagonal.

a(n) is the index of the first occurrence of exactly n consecutive odd numbers in A237271. - Sean A. Irvine, May 03 2026

Links

Table of n, a(n) for n=1..11.

Example

The drawing shows a(1) = 4 and a(2) = 1. Though SRS(4) and SRS(6) share a common boundary of length 2 about the diagonal, SRS(3) and SRS(5) around SRS(4) consist of 2 parts each. The only adjacent neighbor of SRS(1) is SRS(2) and they share a boundary of minimum length 2.
  ._ _ _ _
 6|_ _ _  |_
  ._ _ _|   |_
 4|_ _  |_ _  |
  ._ _|_  | | |
 2|_  | | | | |
 1|_|_|.|_|.|_|.
   1 2   4   6
a(3) = 48 since SRS(48), SRS(49) and SRS(50) have 1, 3 and 3 parts respectively while SRS(47) and SRS(51) have 2 and 4 parts respectively.

Mathematica

(* function partsSRS[ ] is defined in A320511 *)
oddParts[b_] := Select[Map[{#, Length[partsSRS[#]]}&, Range[b]], OddQ[#[[2]]]&]
adjacent[b_] := Split[oddParts[b], #1[[1]]+1==#2[[1]]&]
a395533[b_, c_] := Module[{oPL=adjacent[b], len, list=Table[0, c], i, v}, len=Length[oPL]; For[i=1, i<=len, i++, v=Length[oPL[[i]]]; If[v<=c&&list[[v]]==0, list[[v]]=oPL[[i]][[1, 1]]]]; list]
a395533[2000000, 9] (* long computation time for a(n), n > 7 *)

Crossrefs

Cf. A237593, A240542, A249223, A320511, A237271.

Sequence in context: A372013 A316159 A370415 * A113112 A278578 A395693

Adjacent sequences: A395530 A395531 A395532 * A395534 A395535 A395536

Keyword

nonn,more

Author

Hartmut F. W. Hoft, Apr 27 2026

Extensions

a(10)-a(11) from Sean A. Irvine, May 03 2026

Status

approved