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A280107
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Numbers n with the property that the symmetric representation of sigma(n) has four parts.
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13
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21, 27, 33, 39, 51, 55, 57, 65, 69, 75, 85, 87, 93, 95, 105, 111, 115, 119, 123, 125, 129, 133, 141, 145, 155, 159, 161, 175, 177, 183, 185, 201, 203, 205, 213, 215, 217, 219, 230, 235, 237, 245, 249, 250, 253, 259, 265, 267, 287, 290, 291, 295, 301, 303, 305, 309, 310, 319, 321, 327, 329
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OFFSET
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1,1
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COMMENTS
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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:
(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).
(2) The symmetric representation of sigma(n) consists of four parts.
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)
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LINKS
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EXAMPLE
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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.
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.
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)
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MATHEMATICA
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(* Function a237270[] and row[] are defined in A237270 *)
a280107[m_, n_] := Select[Range[m, n], Length[a237270[#]]==4&]
a280107[1, 329] (* data *)
(* Implementation of the property in the Comment section *)
evenPart[n_] := Module[{f=First[FactorInteger[n]]}, If[First[f]!=2, 1, First[f]^Last[f]]]
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]]
Select[Range[329], fourPartsQ] (* data *)
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CROSSREFS
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First differs from A264102 at a(10).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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