An addition chain for n is a finite increasing sequence of positive
integers whose first element is 1, last element is n, and every element
after the first is the sum of two previous elements in the chain. For
example:
(1 2 4 5 8 10 13)
is an addition for 13.
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A shortest addition chain for n is an addition chain for n with the
smallest possible number of elements. For example,
(1 2 3 6 12 13)
is a shortest addition chain for 13, since there is no addition chain
for 13 with fewer than 6 elements.
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A Brauer chain is an addition chain in which every member after the
first is the sum of the immediately preceeding element and a previous
element (possibly the same element). For example,
(1 2 3 6 7 13)
is a Brauer chain for n.
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A non-Brauer chain for n is an addition chain for n which is not a
Brauer chain. For example
(1 2 4 5 8 13)
is a non-Brauer chain for 13.
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This database records all shortest addition chains for n where
1 <= n <= 1024.
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The file acRead.txt (this file) explains the database format.
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The file acIndx.txt includes summary statistics for the shortest
addition chains for n. The lines are sorted on n, and have the
format
n siz act bct cct
where
n = last element of addition chain
siz = number of elements in shortest addition chain for n
act = number of non-Brauer shortest addition chains for n
bct = number of Brauer shortest addition chains for n
cct = act+bct = total number of shortest addition chains for n
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The file ac****.txt, where **** is a zero-padded number n, records
all shortest addition chains for n. For example, ac0137.txt records
all shortest addition chains for n = 137. Each line describes one of
the shortest addition chains for n. Lines are sorted in lexicographic
order on the chain elements. The format of each line is
a_1 = first element of chain
a_2 = second element of chain
a_3 = third element of chain
...
a_siz = last element of chain
type = a for non-Brauer chain, b for Brauer chain