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A092671
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Numbers n such that there exists a solution to the equation 1 = 1/x_1 + ... + 1/x_k (for any k), 0<x_1<...<x_k=n.
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5
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1, 6, 12, 15, 18, 20, 24, 28, 30, 33, 35, 36, 40, 42, 45, 48, 52, 54, 55, 56, 60, 63, 65, 66, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 115, 117, 119, 120, 126, 130, 132, 133, 135, 136, 138, 140, 143, 144, 145
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| No prime or power of a prime is in this sequence. If n > 1 is in the sequence then all multiples of n are in the sequence. A multiple m*p of a prime p, with all prime factors of m < p, is in the sequence if p is a factor of the numerator of a sum 1/m + 1/x1 +...+ 1/xi, where x1,...xi are distinct integers < m. See A093407 for the least m for each prime p. The Mathematica code uses backtracking to find one solution for a given n. If n is large or not in this sequence, the program will run for a long time. - T. D. Noe (noe(AT)sspectra.com), Mar 30 2004
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REFERENCES
| Harry Ruderman and Paul Erdos, Problem E2427: Bounds of Egyptian fraction partitions of unity, Amer. Math. Monthly, Vol. 81, No. 7 (1974), 780-782.
R. K. Guy, Unsolved Problems in Number Theory, 2nd Ed., New York, Springer-Verlag, 1994, Section D11.
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LINKS
| Toshitaka Suzuki, Table of n, a(n) for n = 1..306
Index entries for sequences related to Egyptian fractions
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MATHEMATICA
| n=55; try3[lev_, s_] := Module[{nmim, nmax, si, i}, AppendTo[soln, 0]; If[lev==1, nmin=2, nmin=1+soln[[ -2]]]; nmax=n-1; Do[If[i<n/2 || !PrimeQ[i], si=s+1/i; If[si==1, soln[[ -1]]=i; Print[soln]; Abort[]]; If[si<1, soln[[ -1]]=i; try3[lev+1, si]]], {i, nmin, nmax}]; soln=Drop[soln, -1]]; soln={n}; try3[1, 1/n] (from T. D. Noe)
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CROSSREFS
| Cf. A092669, A092672.
Cf. A093407 (least m such that m*prime(n) is in this sequence).
Cf. A128253 (primitive elements).
Sequence in context: A114304 A175952 A107487 * A005279 A129512 A196391
Adjacent sequences: A092668 A092669 A092670 * A092672 A092673 A092674
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KEYWORD
| nonn
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AUTHOR
| Max Alekseyev (maxale(AT)gmail.com), Mar 02 2004
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EXTENSIONS
| More terms from T. D. Noe (noe(AT)sspectra.com), Mar 30 2004
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