A117498 Optimal combination of binary and factor methods for finding an addition chain.

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

Offset: 1

Author

Franklin T. Adams-Watters, Mar 22 2006

Keywords

• nonn

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.

For a function f from a finite set X to itself, let I(f) be the number of subsets A of X, which are f-stable in the sense that f(A) is contained in A. Then a(n) is the smallest positive integer m, such that a function f exists on a set with m elements, so that I(f) = n. The f-stable subsets form a finite topology on X, which implies that A137813 is a lower bound. The first index where A137813 is smaller is 23. - Qiaochu Yuan, Jun 27 2024

Examples

			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.
		

Links

• John M. Campbell, A binary version of the Mahler-Popken complexity function, arXiv:2403.20073 [math.NT], 2024. See pp. 5-6. See also Integers (2024) Vol. 24, Art. No. A94. See pp. 5-6.
• Math.Stackexchange, On the number of f-stable subsets, question posted by Alessandrod and answered by Qiaochu Yuan, June 23, 2024.
• Index to sequences related to the complexity of n

Crossrefs

Cf. A003313, A117497, A064097, A137813.

Formula

a(1)=0; a(n) = min(a(n-1)+1, min_{d|n, 1

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