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

%I A320066 #33 Dec 17 2024 08:43:33
%S A320066 63,81,99,117,153,165,195,231,255,273,285,325,345,375,425,435,459,475,
%T A320066 525,561,575,625,627,665,693,725,735,775,805,819,825,875,897,925,975,
%U A320066 1015,1025,1075,1085,1150,1175,1225,1250,1295,1377,1395,1421,1435,1450,1479,1505,1519,1550,1581,1617,1645,1653,1665
%N A320066 Numbers k with the property that the symmetric representation of sigma(k) has five parts.
%C A320066 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
%e A320066 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].
%e A320066 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.
%e A320066 (The diagram of the symmetric representation of sigma(63) = 104 is too large to include.)
%t A320066 (* function a341969 and support functions are defined in A341969, A341970 and A341971 *)
%t A320066 partsSRS[n_] := Length[Select[SplitBy[a341969[n], #!=0&], #[[1]]!=0&]]
%t A320066 a320066[n_] := Select[Range[n], partsSRS[#]==5&]
%t A320066 a320066[1665] (* _Hartmut F. W. Hoft_, Oct 04 2022 *)
%Y A320066 Column 5 of A240062.
%Y A320066 Cf. A000203, A018267, A237270 (the parts), A237271 (number of parts), A174973 (one part), A239929 (two parts), A279102 (three parts), A280107 (four parts).
%Y A320066 Cf. A236104, A237591, A237593, A239663, A239665, A245092, A262626.
%Y A320066 Cf. A341969, A341970, A341971, A357581.
%K A320066 nonn
%O A320066 1,1
%A A320066 _Omar E. Pol_, Oct 05 2018