A351903 a(n) is the smallest number k such that the symmetric representation of sigma(k) has at least one part of size n.

1, 3, 2, 7, 9, 11, 4, 15, 10, 19, 21, 6, 25, 27, 8, 31, 33, 22, 37, 39, 26, 43, 45, 47, 49, 51, 34, 12, 57, 38, 16, 45, 65, 67, 69, 46, 73, 75, 18, 79, 81, 20, 85, 87, 58, 91, 93, 62, 52, 99, 101, 103, 105, 70, 109, 28, 74, 115, 117, 24, 121, 123, 32, 127, 129, 86, 133, 135, 137, 76

Offset

1,2

Comments

Conjecture 1: there are infinitely many pairs of the form a(x) = y; a(y) = x (see examples).

First differs from A351904 at a(11).

From Hartmut F. W. Hoft, Jun 10 2024: (Start)

For numbers less than or equal to a(2^20), (2^k, 2^(k+1) - 1), 0 <= k <= 19, are the only pairs satisfying a(a(x)) = x; the triple (36, 46, 91) is the only one satisfying a(a(a(x))) = x, and there are no proper order 4 quadruples and no order 5 quintuples, apart from fixed point 1.

Conjecture 2: Only the pairs x = 2^k and y = 2^(k+1) - 1, k >= 0, satisfy a(x) = y and a(y) = x.

A repeated number d in this sequence determines a pair of distinct indices u and v such that d = a(u) = a(v). This means that d is the smallest number for which parts of sizes u and v occur in the symmetric representation of sigma(d), SRS(d). There are 5507 such pairs less than a(2^20). (End)

Links

Hartmut F. W. Hoft, Table of n, a(n) for n = 1..1000

Formula

a(n) = min( k : exists 1 <= j <= |SRS(k)|, SRS(k)[j] = n ) where |SRS(k)| denotes the number of parts in SRS(k) and SRS(k)[j] its j-th part. - Hartmut F. W. Hoft, Jun 10 2024

Example

For n = 11 we have that 21 is the smallest number k with at least one part 11 in the symmetric representation of sigma(k), so a(11) = 21.
The symmetric representation of sigma(21) in the first quadrant looks like this:
.
   _ _ _ _ _ _ _ _ _ _ _ 11
  |_ _ _ _ _ _ _ _ _ _ _|
                        |
                        |
                        |_ _ _
                        |_ _  |_ 5
                            |_ _|_
                                | |_ 5
                                |_  |
                                  | |
                                  |_|_ _ _ _ 11
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          |_|
.
For n = 12 we have that 6 is the smallest number k with at least one part 12 in the symmetric representation of sigma(k), so a(12) = 6.
The symmetric representation of sigma(6) in the first quadrant looks like this:
.
   _ _ _ _
  |_ _ _  |_
        |   |_ 12
        |_ _  |
            | |
            | |
            |_|
.
Some pairs of the form a(x) = y; a(y) = x:
   a(2) =  3;   a(3) =  2.
   a(4) =  7;   a(7) =  4.
   a(8) = 15;  a(15) =  8.
  a(16) = 31;  a(31) = 16.
.
The first three repeated terms are 45 = a(23) = a(32), 135 = a(68) = a(104) and 225 = a(113) = a(177), - Hartmut F. W. Hoft, Jun 10 2024

Mathematica

(* a237270[ ] implements improved computing time for A237270 *)
a237591[n_, k_] := Map[Ceiling[(n+1)/#-(#+1)/2]-Ceiling[(n+1)/(#+1)-(#+2)/2]&, Range[k]]
a249223[n_, k_] := FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, k]]
row[n_] := Floor[(Sqrt[8*n+1]-1)/2]
a237270[n_] := Module[{lw=Transpose[{a237591[n, row[n]], a249223[n, row[n]]}], diag, sL, pL}, diag=Last[lw][[2]]; sL=Split[lw, #[[2]]!=0&]; pL=Map[Apply[Dot, Transpose[#]]&, Select[sL, #[[1, 2]]!=0&]]; If[diag==0, Join[pL, Reverse[pL]], If[Length[pL]>1, Join[Most[pL], {2Last[pL]-diag}, Reverse[Most[pL]]], 2pL-diag]]]
a351903[n_] := Module[{list=Table[0, n], count0=n, i=1}, While[count0>0, Map[If[list[[#]]==0, list[[#]]=i; count0--]&, Select[a237270[i], #<=n&]]; i++]; list]
a351903[70] (* Hartmut F. W. Hoft, Jun 10 2024 *)

Crossrefs

Cf. A351904 (Analog for subparts).

Cf. A000079, A000203, A000225, A196020, A235791, A236104, A237270 (parts), A237271, A237591, A237593, A239660, A239663, A239931, A239932, A239933, A239934, A241558, A245092, A262626.

Sequence in context: A374611 A118966 A351904 * A359973 A348535 A018891

Adjacent sequences: A351900 A351901 A351902 * A351904 A351905 A351906

Keyword

nonn

Author

Omar E. Pol, Feb 25 2022

Extensions

a(17)-a(70) from Hartmut F. W. Hoft, Jun 10 2024

Status

approved