Egyptian Fractions

[Franklin T. Adams-Watters]

The existing sequences in the OEIS relating to Egyptian fractions almost all relate to expressing 1 as an Egyptian fraction.  We really should add some sequences relating to the optimal representation of fractions between zero and one as an Egyptian fraction.

It seems to me that there are four very reasonable definitions of "optimal" that we should consider:

A Minimum number of terms.
B Minimum maximum term.
C Minimum sum of terms.
D Minimum least common multiple of terms (when you add the fractions in one go, this will be the denominator before reduction to lowest terms).

One can also look at D', which is D divided by the denominator.

There are also, it seems to me, three logical presentations for each of these:

1 A simple triangular representation, with fractions form 1/n to n/n in each row.
2 A reduced table, with only the fractions in lowest terms in each row.
3 A simple sequence, with the maximum value from each row.

Here is the start of table A1 (through 22), with the values not in A2 followed by an asterisk:

1
1 1*
1 2 1*
1 1* 2 1*
1 2 2 3 1*
1 1* 1* 2* 2 1*
1 2 3 2 3 3 1*
1 1* 2 1* 2 2* 3 1*
1 2 1* 2 2 2* 3 3 1*
1 1* 2 2* 1* 2* 2 3* 3 1*
1 2 2 2 3 2 3 4 4 4 1*
1 1* 1* 1* 2 1* 2 2* 2* 2* 3 1*
1 2 3 3 3 3 2 3 3 3 3 4 1*
1 1* 2 2* 2 3* 1* 2* 2 3* 3 3* 4 1*
1 2 1* 2 1* 2* 3 2 2* 2* 3 3* 3 3 1*
1 1* 2 1* 2 2* 3 1* 2 2* 3 2* 3 3* 4 1*
1 2 2 3 3 2 3 4 2 3 3 3 3 4 4 5 1*
1 1* 1* 2* 2 1* 2 2* 1* 2* 2 2* 3 3* 2* 3* 3 1*
1 2 3 2 2 3 3 3 4 2 3 3 3 4 4 3 4 4 1*
1 1* 2 1* 1* 2* 2 2* 2 1* 2 2* 3 2* 2* 3* 3 3* 3 1*
1 2 1* 2 2 2* 1* 2 3* 2 2 2* 3 2* 3* 3 3 3* 3 4 1*
1 1* 2 2* 3 2* 3 2* 3 3* 1* 2* 2 3* 3 4* 3 4* 3 4* 3 1*

(These values were mechanically generated, but transcribed by hand; verification would be apprecieated.)

This gives us for A3 (through n=32):

1 1 2 2 3 2 3 3 3 3 4 3 4 4 3 4 5 3 4 3 4 5 3 4 4 4 4 5 4 5 4

For B2, calculating by hand, I get:

1
2
3 6
4 4
5 15 10 10
6 3
7 21 21 14 14 21
8 8 8 8
9 18 9 18 9 18
10 10 5 15
11 44 44 33 33 22 22 44 44 33
12 6 4 6
13 65 39 39 65 52 26 26 39 39 52 65
14 14 21 7 21 21
15 20 10 10 5 15 6 10
16 16 16 16 16 16 16 16
17 68 68 85 68 51 51 85 34 34 68 68 68 68 51 51
18 9 18 9 18 9
19 114 95 57 57 76 114 95 76 38 38 114 57 57 76 76 95 76

And for C2, I get:

1
2
3 8
4 6
5 18 12 17
6 5
7 32 39 16 23 36
8 12 10 14 18 12 20 17 23
9 18 12 20 17 23
10 15 7 20
11 72 48 36 47 24 35 95 72 60
12 10 7 12
13 98 71 82 95 101 28 41 47 58 71 96
14 21 34 9 34 41
15 32 16 21 8 23 13 15
16 24 20 28 18 26 22 30
17 131 108 125 106 54 71 120 36 53 165 144 74 91 90 107
18 15 21 11 26 14
19 185 198 100 80 99 155 174 180 40 59 162 65 84 120 119 138 145

Verification and extension of these values would also be appreciated.

I have not calculated any values for optimization D.

I will follow this up with a separate note on calculation techniques for these values.

-- 
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645


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