sum of unit fractions

[Franklin T. Adams-Watters]

David's "good" numbers are already in the OEIS: A092671.

"David Wilson" <davidwwilson at comcast.net> wrote:
>At any rate, I had an interesting idea (I think).
>
>Let m be good if it is the largest element of a set whose reciprocals
>add to an integer.  More precisely, letting M(S) be the maximum element
>of S, m is good if there exists S with R(S) integer and M(S) = m.
>
>The sets
>
>    S = { 1 }
>    S = { 1, 2, 3, 6 }
>    S = { 1, 2, 3, 4, 5, 6, 8, 9, 10, 15, 18, 20, 24 }
>
>demonstrate that 1, 6 and 24 are good.
>
>Primes, however, are bad.  For suppose prime p is good.  Then there exists
>S with R(S) integer and M(S) = p. We must then have S_p = { p }, giving
>R(S_p) = 1/p, and we cannot eliminate p from the denominator.  The
>Above Theorem then implies R(S) is not an integer.
>
>The primes are not the only bad numbers, for instance, 10 is also bad.
>For suppose 10 is good.  Then S exists with R(S) integer with M(S) = 10.
>Then S_5 = { 5, 10 } or { 10 }, giving R(S_5) = 3/10 or 1/10.  Either
>way, 5 appears in the denominator, and the Above Theorem implies that
>R(S) is not an integer.
>
>These observations beg the questions:
>
>(1) Which integers are good?  This has the potential for an OEIS
>sequence.
>
>(2) Is the Above Theorem sufficient to show an integer is bad?
>
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-- 
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645


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