

A117498


Optimal combination of binary and factor methods for finding an addition chain.


1



0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 5, 4, 5, 5, 6, 5, 6, 6, 7, 5, 6, 6, 6, 6, 7, 6, 7, 5, 6, 6, 7, 6, 7, 7, 7, 6, 7, 7, 8, 7, 7, 8, 9, 6, 7, 7, 7, 7, 8, 7, 8, 7, 8, 8, 9, 7, 8, 8, 8, 6, 7, 7, 8, 7, 8, 8, 9, 7, 8, 8, 8, 8, 9, 8, 9, 7, 8, 8, 9, 8, 8, 9, 9, 8, 9, 8, 9, 9, 9, 10, 9, 7, 8, 8, 8, 8, 9, 8, 9, 8, 9
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OFFSET

1,3


COMMENTS

This is an upper bound for both addition chains (A003313) and A117497. The first few values where A003313 is smaller are 23,43,46,47,59. The first few values where A117497 is smaller are 77,143,154,172,173. The first few values where both are smaller are 77,154,172,173,203.


LINKS

Table of n, a(n) for n=1..105.
Index to sequences related to the complexity of n


FORMULA

a(1)=0; a(n) = min(a(n1)+1, min_{dn, 1<d<n} a(d)+a(n/d)). If n is prime, this reduces to a(n) = a(n1)+1.


EXAMPLE

a(33)=6 because 6 = 1+a(32) < a(3)+a(11) = 2+5. a(36) = min(a(35)+1, a(2)+a(18), a(3)+a(12), a(4)+a(9), a(6)+a(6)) = min(1+7, 1+5, 2+4, 2+4, 3+3) = 6.


CROSSREFS

Cf. A003313, A117497, A064097.
Sequence in context: A003313 A277608 A117497 * A064097 A014701 A207034
Adjacent sequences: A117495 A117496 A117497 * A117499 A117500 A117501


KEYWORD

nonn


AUTHOR

Franklin T. AdamsWatters, Mar 22 2006


STATUS

approved



