sum of unit fractions

Paul D. Hanna pauldhanna at juno.com
Thu Dec 9 03:11:31 CET 2004

```     A very interesting development.
Of course, other interesting sequences arise from this besides
{1,6,24,65,...}.

Assuming that the sets of unit fractions that sum to n are accumulative,
so that the (n+1)-th set contains the n-th set,
then we can separate these into subsets of unit fractions that sum to
unity.

Then if we examine the distinct sets of unit fractions that sum to unity,

we get the following for each set (if any one of these could be
predicted,
it would be a big help in finding the unit fractions in each set).

Initial terms:
{1,2,5,7,17,29,...}

Number of terms:
{1,4,8,24,73,182,...}

Sum of terms:
{1,15,109,835,7524,48639,...}

And the distinct sets of unit fractions that sum to unity are:
{1}

{2,3,4,6}

{5,8,9,10,15,18,20,24}

{7,11,12,13,14,16,22,26,27,28,30,33,35,36,40,42,
45,48,52,54,56,60,63,65}

{17,19,21,23,25,32,34,38,39,44,50,51,55,58,62,66,68,
69,70,72,75,76,77,78,80,81,84,85,87,88,90,91,92,93,
95,96,99,102,104,105,108,110,112,114,115,116,117,126,
130,133,136,138,140,143,144,145,150,152,153,154,155,
156,161,162,165,168,170,171,174,175,176,180,184}

{29,31,37,41,43,46,47,49,53,57,61,64,67,74,82,86,94,
98,100,106,111,119,120,121,122,123,124,128,129,132,134,
135,141,147,148,159,160,164,177,182,183,185,186,187,188,
189,190,192,195,196,198,200,201,203,205,207,208,209,210,
212,215,216,217,220,221,222,224,225,228,230,231,232,234,
238,240,242,245,246,247,248,250,252,253,255,258,259,260,
261,264,266,268,270,272,273,275,276,280,282,285,286,287,
288,290,294,295,297,299,300,301,304,305,306,308,310,312,
315,319,320,322,323,324,325,328,329,330,336,340,341,342,
344,345,348,350,351,352,354,357,363,364,370,372,374,375,
376,377,378,384,387,390,391,396,402,405,406,407,408,413,
414,416,418,420,424,425,429,430,432,434,435,437,442,444,
448,450,451,455,456,460,462,465,468,469,475}

But I am sure that you have already thought of these as well.
Paul

---------------------------------------------------------------
On Wed, 08 Dec 2004 00:08:06 +0000 hv at crypt.org writes:
> Define a(n) as the least integer k such that there is a sum of
> distinct unit
> fractions equal to _n_ of which the greatest denominator is k.
>
[...]
>
> There is no sequence currently in the database matching 1, 6, 24, 65.
>
> Hugo van der Sanden

```