[seqfan] A083207 On an observation of Frank Buss.

peter.luschny peter.luschny at googlemail.com
Thu Jul 8 20:58:38 CEST 2010

Recently numbers n whose divisors can be partitioned into two
disjoint sets whose sums are both sigma(n)/2, which were introduced by
Reinhard Zumkeller in 2003, got attention in the mathematical literature,
in a discussion on the Usenet de.sci.mathematik and by T. D. Noe on OEIS.

Frank Buss examined the difference of consecutive Zumkeller
numbers up to 27188 and observed that greatest distance in this
region is 12. (A plot can be seen here [1]).

This observation was extended by T. D. Noe. He writes: "The 229026
Zumkeller numbers less than 10^6 have a maximum difference of 12.
This leads to the conjecture that any 12 consecutive numbers have
at least one Zumkeller number."

Any comments on this conjecture are highly welcome.

Cheers, Peter

[1] http://www.luschny.de/math/seq/ZumkellerNumbers.html
[2] http://www.research.att.com/~njas/sequences/?q=id%3AA083207

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