A320511 - OEIS

%I #21 Dec 17 2024 08:42:58

%S 147,171,189,207,243,261,275,279,297,333,351,363,369,387,423,429,465,

%T 477,507,531,549,555,595,603,605,615,639,645,657,663,705,711,715,741,

%U 747,795,801,833,845,867,873,885,909,915,927,931,935,963,969,981,1005,1017,1045,1065,1071,1083,1095,1105,1127

%N Numbers k with the property that the symmetric representation of sigma(k) has six parts.

%C Those numbers in this sequence with only parts of width 1 in their symmetric representation of sigma form column 6 in the table of A357581. - _Hartmut F. W. Hoft_, Oct 04 2022

%e 147 is in the sequence because the 147th row of A237593 is [74, 25, 13, 8, 5, 4, 4, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 4, 4, 5, 8, 13, 25, 74], and the 146th row of the same triangle is [74, 25, 12, 8, 6, 4, 3, 2, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 4, 6, 8, 12, 25, 74], therefore between both symmetric Dyck paths there are six parts: [74, 26, 14, 14, 26, 74].

%e Note that the sum of these parts is 74 + 26 + 14 + 14 + 26 + 74 = 228, equaling the sum of the divisors of 147: 1 + 3 + 7 + 21 + 49 + 147 = 228.

%e (The diagram of the symmetric representation of sigma(147) = 228 is too large to include.)

%t (* function a341969 and support functions are defined in A341969, A341970 and A341971 *)

%t partsSRS[n_] := Length[Select[SplitBy[a341969[n], #!=0&], #[[1]]!=0&]]

%t a320511[n_] := Select[Range[n], partsSRS[#]==6&]

%t a320511[1127] (* _Hartmut F. W. Hoft_, Oct 04 2022 *)

%Y Column 6 of A240062.

%Y Cf. A237270 (the parts), A237271 (number of parts), A174973 (one part), A239929 (two parts), A279102 (three parts), A280107 (four parts), A320066 (five parts).

%Y Cf. A000203, A018303, A196020, A235791, A236104, A237048, A237591, A237593, A239663, A239665, A245092, A262626, A296508.

%Y Cf. A341969, A341970, A341971, A357581.

%K nonn

%O 1,1

%A _Omar E. Pol_, Oct 14 2018