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A117498
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Optimal combination of binary and factor methods for finding an addition chain.
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1
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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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.
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FORMULA
| a(1)=0; a(n) = min(a(n-1)+1, min_{d|n, 1<d<n} a(d)+a(n/d)). If n is prime, this reduces to a(n) = a(n-1)+1.
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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.
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CROSSREFS
| Cf. A003313, A117497, A064097.
Sequence in context: A137813 A003313 A117497 * A064097 A014701 A056239
Adjacent sequences: A117495 A117496 A117497 * A117499 A117500 A117501
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KEYWORD
| nonn
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AUTHOR
| Frank Adams-Watters (FrankTAW(AT)Netscape.net), Mar 22 2006
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