A265999 -id:A265999 - cp's OEIS frontend

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

Omar E. Pol, Dec 19 2015

nonn

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

			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)

Complement of A265999.
Cf. A000203, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A245092, A262626.
Cf. A239663, A251820, A377654.

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

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

cp's OEIS Frontend

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

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