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Numbers k such that the symmetric representation of sigma(k) has at least two parts of distinct size.
2

%I #35 Jan 23 2025 15:36:13

%S 9,21,25,27,33,35,39,45,49,50,51,55,57,63,65,69,70,75,77,81,85,87,91,

%T 93,95,98,99,105,110,111,115,117,119,121,123,125,129,130,133,135,141,

%U 143,145,147,153,154,155,159,161,165,169,170,171,175,177,182,183,185,187,189,190,195

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

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

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

%C 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

%e The symmetric representation of sigma(9) = 13 in the first quadrant looks like this:

%e y

%e .

%e ._ _ _ _ _ 5

%e |_ _ _ _ _|

%e . |_ _ 3

%e . |_ |

%e . |_|_ _ 5

%e . | |

%e . | |

%e . | |

%e . | |

%e . . . . . . . . |_| . . x

%e .

%e 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.

%e From _Hartmut F. W. Hoft_, Jan 11 2025: (Start)

%e SRS(a(1)) = SRS(A239663(3)) = SRS(9) = { 5, 3, 5 } is the smallest with 2 parts of distinct sizes.

%e SRS(a(14)) = SRS(A239663(5)) = SRS(63) = { 32, 12, 16, 12, 32 } is the smallest with 3 parts of distinct sizes.

%e 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)

%t (* Function partsSRS[ ] is defined in A377654 *)

%t a266000[n_] := Select[Range[n], Length[Union[partsSRS[#]]]>=2&]

%t a266000[200] (* _Hartmut F. W. Hoft_, Jan 11 2025 *)

%Y Complement of A265999.

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

%Y Cf. A239663, A251820, A377654.

%K nonn,changed

%O 1,1

%A _Omar E. Pol_, Dec 19 2015

%E Extended from a(37) to a(62) by _Hartmut F. W. Hoft_, Jan 11 2025