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%I A280107 #30 Oct 04 2018 06:54:35
%S A280107 21,27,33,39,51,55,57,65,69,75,85,87,93,95,105,111,115,119,123,125,
%T A280107 129,133,141,145,155,159,161,175,177,183,185,201,203,205,213,215,217,
%U A280107 219,230,235,237,245,249,250,253,259,265,267,287,290,291,295,301,303,305,309,310,319,321,327,329
%N A280107 Numbers n with the property that the symmetric representation of sigma(n) has four parts.
%C A280107 From _Hartmut F. W. Hoft_, Jan 27 2018: (Start)
%C A280107 Let  n = 2^k * t where k >= 0 and t is odd, and let D be the set of divisors of t less than r(n) = floor((sqrt(8*n+1) - 1)/2). The following statements are equivalent:
%C A280107 (1)  There is exactly one d in D such that 2^(k+1) * d < e where e in D is the next odd divisor larger than d, and the largest divisor f in D satisfies 2^(k+1) * f <= r(n).
%C A280107 (2)  The symmetric representation of sigma(n) consists of four parts.
%C A280107 The property says that the first part of the symmetric representation of n consists of the first 2^(k+1) * d - 1 legs and that the second part starts with leg e and ends with leg 2^(k+1) * f - 1 before or at the middle of the Dyck path (see A237048 and A249223) on the diagonal. Together with their symmetric pair they form the four parts. (End)
%e A280107 a(1) = 21 because it is the smallest number n whose symmetric representation of sigma(n) has four parts. Note that the sum of the parts is 11 + 5 + 5 + 11 = 32, equaling the sum of the divisors of 21: aigma(21) = 1 + 3 + 7 + 21 = 32.
%e A280107 From _Hartmut F. W. Hoft_, Jan 27 2018: (Start)
%e A280107 230 = 2*5*23 is the first even number since 2^2 < 5, 2^2 * 5 < 23, and row 230 in A237048 has 20 entries with 1's in positions 1, 4, 5, and 20.
%e A280107 Prime number 3 can be a factor for an even number in this sequence as 12246=2*3*13*157 demonstrates with the four parts 12252, 1020, 1020, and 12252 in the symmetric representation of sigma(12246) defined by 1's in positions 1, 3, 4, 12, 13, 39, 52, 156 in row 12246 of A237048; each of the four parts has maximum width 2 and the two central parts meet on the diagonal at 8492. (End)
%t A280107 (* Function a237270[] and row[] are defined in A237270 *)
%t A280107 a280107[m_, n_] := Select[Range[m, n], Length[a237270[#]]==4&]
%t A280107 a280107[1, 329] (* data *)
%t A280107 (* Implementation of the property in the Comment section *)
%t A280107 evenPart[n_] := Module[{f=First[FactorInteger[n]]}, If[First[f]!=2, 1, First[f]^Last[f]]]
%t A280107 fourPartsQ[n_] := Module[{e=evenPart[n], oddPart, r=row[n], dL}, oddPart=n/evenPart[n]; dL=Select[Divisors[oddPart], #<r&]; If[Length[dL]>1, 2*e*Last[dL]<=r && Length[Select[2*e*Most[dL]-Rest[dL], #<0&]==1, False]]
%t A280107 Select[Range[329], fourPartsQ] (* data *)
%t A280107 (* _Hartmut F. W. Hoft_, Jan 27 2018 *)
%Y A280107 Column 4 of A240062.
%Y A280107 First differs from A264102 at a(10).
%Y A280107 Numbers n such that the symmetric representation of sigma(n) has k parts (k = 1..4): A174973 = A238443, A239929, A279102, this sequence.
%Y A280107 Cf. A237270, A237271 (number of parts), A237591, A237593, A239665, A245092, A262626.
%Y A280107 Cf. A237048, A249223. - _Hartmut F. W. Hoft_, Jan 27 2018
%K A280107 nonn
%O A280107 1,1
%A A280107 _Omar E. Pol_, Dec 27 2016

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