A266000 Numbers k such that the symmetric representation of sigma(k) has at least two parts of distinct size.

9, 21, 25, 27, 33, 35, 39, 45, 49, 50, 51, 55, 57, 63, 65, 69, 70, 75, 77, 81, 85, 87, 91, 93, 95, 98, 99, 105, 110, 111, 115, 117, 119, 121, 123, 125, 129, 130, 133, 135, 141, 143, 145, 147, 153, 154, 155, 159, 161, 165, 169, 170, 171, 175, 177, 182, 183, 185, 187, 189, 190, 195

Offset

1,1

Comments

In other words: numbers k such that the symmetric representation of sigma(k) has at least two parts with distinct number of cells.

For more information about the symmetric representation of sigma see A237270 and A237593.

When the symmetric representation of sigma of m, SRS(m), consists of 2n-1 or 2n parts, n>=1, then at most n parts can be of distinct sizes. For the published terms in A239663, SRS(A239663(n)) consists of n parts representing ceiling(n/2) parts of distinct sizes, n>=1. Only two numbers m are known, 15 and 5950 in A251820, for which SRS(m) consists of n parts of less than ceiling(n/2) distinct sizes. - Hartmut F. W. Hoft, Jan 11 2025

Links

Table of n, a(n) for n=1..62.

Example

The symmetric representation of sigma(9) = 13 in the first quadrant looks like this:
y
.
._ _ _ _ _ 5
|_ _ _ _ _|
.         |_ _ 3
.         |_  |
.           |_|_ _ 5
.               | |
.               | |
.               | |
.               | |
. . . . . . . . |_| . . x
.
There are three parts: 5 + 3 + 5 = 13, so 9 is in the sequence because the structure contains at least two parts of distinct size.
From Hartmut F. W. Hoft, Jan 11 2025: (Start)
SRS(a(1)) = SRS(A239663(3)) = SRS(9) = { 5, 3, 5 } is the smallest with 2 parts of distinct sizes.
SRS(a(14)) = SRS(A239663(5)) = SRS(63) = { 32, 12, 16, 12, 32 } is the smallest with 3 parts of distinct sizes.
SRS(a(127)) = SRS(A239663(7)) = SRS(357) = { 179, 61, 29, 38, 29, 61, 179 } is the smallest with 4 parts of distinct sizes. (End)

Mathematica

(* Function partsSRS[ ] is defined in A377654 *)
a266000[n_] := Select[Range[n], Length[Union[partsSRS[#]]]>=2&]
a266000[200] (* Hartmut F. W. Hoft, Jan 11 2025 *)

Crossrefs

Complement of A265999.

Cf. A000203, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A245092, A262626.

Cf. A239663, A251820, A377654.

Sequence in context: A111171 A317789 A333039 * A067887 A345330 A174870

Adjacent sequences: A265997 A265998 A265999 * A266001 A266002 A266003

Keyword

nonn

Author

Omar E. Pol, Dec 19 2015

Extensions

Extended from a(37) to a(62) by Hartmut F. W. Hoft, Jan 11 2025

Status

approved