A320066 Numbers k with the property that the symmetric representation of sigma(k) has five parts.

63, 81, 99, 117, 153, 165, 195, 231, 255, 273, 285, 325, 345, 375, 425, 435, 459, 475, 525, 561, 575, 625, 627, 665, 693, 725, 735, 775, 805, 819, 825, 875, 897, 925, 975, 1015, 1025, 1075, 1085, 1150, 1175, 1225, 1250, 1295, 1377, 1395, 1421, 1435, 1450, 1479, 1505, 1519, 1550, 1581, 1617, 1645, 1653, 1665

Offset: 1

Author

Omar E. Pol, Oct 05 2018

Keywords

• nonn

Comments

Those numbers in this sequence with only parts of width 1 in their symmetric representation of sigma form column 5 in the table of A357581. - Hartmut F. W. Hoft, Oct 04 2022

Examples

			63 is in the sequence because the 63rd row of A237593 is [32, 11, 6, 4, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 2, 4, 6, 11, 32], and the 62nd row of the same triangle is [32, 11, 5, 4, 3, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 3, 4, 5, 11, 32], therefore between both symmetric Dyck paths there are five parts: [32, 12, 16, 12, 32].
The sums of these parts is 32 + 12 + 16 + 12 + 32 = 104, equaling the sum of the divisors of 63: 1 + 3 + 7 + 9 + 21 + 63 = 104.
(The diagram of the symmetric representation of sigma(63) = 104 is too large to include.)
		

Crossrefs

Column 5 of A240062.

Cf. A000203, A018267, A237270 (the parts), A237271 (number of parts), A174973 (one part), A239929 (two parts), A279102 (three parts), A280107 (four parts).

Cf. A236104, A237591, A237593, A239663, A239665, A245092, A262626.

Cf. A341969, A341970, A341971, A357581.

Programs

Mathematica

(* function a341969 and support functions are defined in A341969, A341970 and A341971 *)
partsSRS[n_] := Length[Select[SplitBy[a341969[n], #!=0&], #[[1]]!=0&]]
a320066[n_] := Select[Range[n], partsSRS[#]==5&]
a320066[1665] (* Hartmut F. W. Hoft, Oct 04 2022 *)