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A092671 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. 10
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; text; internal format)
OFFSET

1,2

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, Mar 30 2004

Conjecture (verified through n=2*10^5): For any n>1, let P be the largest divisor of n that is either a prime (p) or prime power (p^e, where e>1), and let m=n/P. Then n is in the sequence iff p is a factor of the numerator of a sum 1/m + 1/x_1 +...+ 1/x_i, where x_1,...,x_i are distinct integers < m. - Jon E. Schoenfield, Apr 06 2014

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, 2nd Ed., New York, Springer-Verlag, 1994, Section D11.

LINKS

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 306 terms from Toshitaka Suzuki)

Harry Ruderman and Paul Erdős, Problem E2427: Bounds of Egyptian fraction partitions of unity, Amer. Math. Monthly, Vol. 81, No. 7 (1974), 780-782.

Index entries for sequences related to Egyptian fractions

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] (* T. D. Noe *)

CROSSREFS

Cf. A092669, A092672.

Cf. A093407 (least m such that m*prime(n) is in this sequence).

Cf. A128253 (primitive elements).

Sequence in context: A208770 A219095 A107487 * A239658 A005279 A129512

Adjacent sequences:  A092668 A092669 A092670 * A092672 A092673 A092674

KEYWORD

nonn

AUTHOR

Max Alekseyev, Mar 02 2004

EXTENSIONS

More terms from T. D. Noe, Mar 30 2004

STATUS

approved

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Last modified March 26 23:19 EDT 2017. Contains 284141 sequences.