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An
Egyptian representation of a
real number is a sum of positive (usually) distinct
unit fractions equal to
. The (unique)
greedy Egyptian representation uses the
greedy algorithm for Egyptian representation.
Greedy Egyptian representation of 1 in distinct unit fractions less than 1

1 = + + + + + + + 1  113423713055421844361000443  + ⋯ 
The denominators are given by the quadratic recurrence

a (1) = 2; a (n) = a (n − 1) [a (n − 1) − 1] + 1 = a (n − 1) 2 − a (n − 1) + 1, n ≥ 2. 
and by the formula (which shows that it is an infinite coprime sequence)

a (n) = 1 + a (i), n ≥ 1, 
where for
we get
1 + (empty product, i.e.
1) = 2.
A000058 Sylvester’s sequence:
a (n + 1) = a (n) 2 − a (n) + 1 
, with
. (Greedy Egyptian representation of
1.)

{2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, 12864938683278671740537145998360961546653259485195807, ...}
Sylvester’s sequence, except for the first term (i.e.
2), is a subsequence of the
central polygonal numbers (
quasioblong numbers), where we start with the second (since
) term (i.e.
3), then use the value of the second term as the index (
3) of the third term (i.e.
7), then use the value of the third term as the index (
7) of the fourth term (i.e.
43), then use the value of the fourth term as the index (
43) of the fourth term (i.e.
1807), and so on...
A002061 Central polygonal numbers:
. (Quasioblong numbers:
.)

{1, 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, 241, 273, 307, 343, 381, 421, 463, 507, 553, 601, 651, 703, 757, 813, 871, 931, 993, 1057, 1123, 1191, 1261, 1333, 1407, 1483, 1561, 1641, 1723, 1807, ...}
Greedy Egyptian representation of positive rational numbers less than 1
Greedy Egyptian representation

Denominators of distinct unit fractions of the greedy Egyptian representation of positive rational numbers less than 1


{2}


{3}


{2, 6}


{4}


{2, 4}


{5}


{3, 15}


{2, 10}


{2, 4, 20}


{6}


{2, 3}


{7}


{4, ...}


{3, ...}


{2, ...}


{2, ...}


{2, ...}


{8}


{3, ...}


{2, ...}


{2, ...}


{9}


{5, ...}


{3, ...}


{2, ...}


{2, ...}


{2, ...}


{10}


{4, ...}


{2, ...}


{2, ...}

Greedy algorithm for Egyptian representation
Fibonacci’s algorithm for the greedy Egyptian representation
 ${\begin{array}{l}\displaystyle {E(r)=\sum _{i\geq 1,\,E_{i}>0}E_{i}}\end{array}}$
of a rational number
is

r0 = r ; ri = ri − 1 − Ei , i ≥ 1, 
where

where
is the
ceiling function.
Greedy signed Egyptian representation
A
signed Egyptian representation of a
real number is a sum of negative or positive (usually) distinct
unit fractions equal to
. The (unique)
greedy signed Egyptian representation uses the
greedy algorithm for signed Egyptian representation.
Greedy signed Egyptian representation of positive rational numbers less than 1
Greedy signed Egyptian representation

Signed denominators of distinct signed unit fractions of the greedy signed Egyptian representation of positive rational numbers less than 1


{2}


{3}


{2, 6}


{4}


{2, 4}


{5}


{2, −10}


{2, 10}


{2, ...}


{6}


{2, 3}


{7}


{4, 28}


{2, −14}


{2, 14}


{2, ...}


{2, ...}


{8}


{2, −8}


{2, 8}


{2, ...}


{9}


{4, −36}


{2, −18}


{2, 18}


{2, ...}


{2, ...}


{10}


{3, −30}


{2, 5}


{2, ...}

Greedy algorithm for signed Egyptian representation
***** NEEDS DOUBLECHECKING *****
The greedy algorithm for the signed Egyptian representation
 ${\begin{array}{l}\displaystyle {S=\sum _{i\geq 1,\,S_{i}>0}(1)^{H(r_{i1})}~\lfloor S_{i}\rfloor ,}\end{array}}$
where
is the
Heaviside step function, of a rational number
is

r0 = r ; ri = ri − 1 − Si , i ≥ 1, 

where
is the
nearest integer function.
Sequences
Denominators of distinct unit fractions (less than 1) of the greedy Egyptian representation of rational numbers (less than 1) ordered by increasing denominators, then by increasing coprime numerators of those rational numbers
 {{2}, {3}, {2, 6}, {4}, {2, 4}, {5}, {3, 15}, {2, 10}, {2, 4, 20}, {6}, {2, 3}, {7}, ...}
whose concatenation gives the following sequence.
A??????

{2, 3, 2, 6, 2, 4, 5, 3, 15, 2, 10, 2, 4, 20, 6, 2, 3, 7, ...}
See also