

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
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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, SpringerVerlag, 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 ErdÅ‘s, Problem E2427: Bounds of Egyptian fraction partitions of unity, Amer. Math. Monthly, Vol. 81, No. 7 (1974), 780782.
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=n1; 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



