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A003313 Length of shortest addition chain for n.
(Formerly M0255)
45
0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 5, 4, 5, 5, 6, 5, 6, 6, 6, 5, 6, 6, 6, 6, 7, 6, 7, 5, 6, 6, 7, 6, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 8, 6, 7, 7, 7, 7, 8, 7, 8, 7, 8, 8, 8, 7, 8, 8, 8, 6, 7, 7, 8, 7, 8, 8, 9, 7, 8, 8, 8, 8, 8, 8, 9, 7, 8, 8, 8, 8, 8, 8, 9, 8, 9, 8, 9, 8, 9, 9, 9, 7, 8, 8, 8, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Equivalently, minimal number of multiplications required to compute n-th power.

Equivalently, for n>1, the minimum number of points m(n) needed to make a topology having n open sets, as shown in Table 1, p.2 of Ragnarsson and Tenner. [Jonathan Vos Post, Oct 08 2008]

REFERENCES

Bahig, Hatem M.; El-Zahar, Mohamed H.; Nakamula, Ken; Some results for some conjectures in addition chains, in Combinatorics, computability and logic, pp. 47-54, Springer Ser. Discrete Math. Theor. Comput. Sci., Springer, London, 2001.

Bergeron, F.; Berstel, J.; Brlek, S.; Duboc, C.; Addition chains using continued fractions. J. Algorithms 10 (1989), 403-412.

D. Bleichenbacher and A. Flammenkamp, An Efficient Algorithm for Computing Shortest Addition Chains, Preprint, 1997.

Brauer, Alfred, On addition chains. Bull. Amer. Math. Soc. 45, (1939). 736-739.

Downey, Peter; Leong, Benton; Sethi, Ravi; Computing sequences with addition chains. SIAM J. Comput. 10 (1981), 638-646.

Elia, M. and Neri, F.; A note on addition chains and some related conjectures, in Sequences (Naples/Positano, 1988), pp. 166-181, Springer, New York, 1990.

P. Erdos, Remarks on number theory. III. On addition chains. Acta Arith. 6 1960 77-81.

A. Flammenkamp, Drei Beitraege zur diskreten Mathematik: Additionsketten, No-Three-in-Line-Problem, Sociable Numbers, Diplomarbeit, Bielefeld 1991.

Gashkov, S. B. and Kochergin, V. V.; On addition chains of vectors, gate circuits and the complexity of computations of powers [translation of Metody Diskret. Anal. No. 52 (1992), 22-40, 119-120; 1265027], Siberian Adv. Math. 4 (1994), 1-16.

Gioia, A. A.; Subba Rao, M. V.; Sugunamma, M.; The Scholz-Brauer problem in addition chains. Duke Math. J. 29 1962 481-487.

Gioia, A. A. and Subbarao, M. V., The Scholz-Brauer problem in addition chains, II, in Proceedings of the Eighth Manitoba Conference on Numerical Mathematics and Computing (Univ. Manitoba, Winnipeg, Man., 1978), pp. 251-274, Congress. Numer., XXII, Utilitas Math., Winnipeg, Man., 1979.

Graham, R. L.; Yao, A. C. C.; Yao, F. F., Addition chains with multiplicative cost. Discrete Math. 23 (1978), 115-119.

D. E. Knuth, The Art of Computer Programming, vol. 2, Seminumerical Algorithms, 2nd ed., Fig. 14 on page 403; 3rd edition, 1998, p. 465.

D. E. Knuth, website, further updates to Vol. 2 of TAOCP.

McCarthy, D. P., Effect of improved multiplication efficiency on exponentiation algorithms derived from addition chains. Math. Comp. 46 (1986), 603-608.

Olivos, Jorge, On vectorial addition chains. J. Algorithms 2 (1981), 13-21.

Schoenhage, Arnold, A lower bound for the length of addition chains. Theor. Comput. Sci. 1 (1975), 1-12.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Thurber, Edward G. The Scholz-Brauer problem on addition chains. Pacific J. Math. 49 (1973), 229-242.

Thurber, Edward G. On addition chains ... and lower bounds for c(r). Duke Math. J. 40 (1973), 907-913.

Thurber, Edward G., Addition chains and solutions of l(2n)=l(n) and l(2^n-1)=n+l(n)-1. Discrete Math. 16 (1976), 279-289.

Thurber, Edward G., Addition chains-an erratic sequence. Discrete Math. 122 (1993), 287-305.

Thurber, Edward G., Efficient generation of minimal length addition chains. SIAM J. Comput. 28 (1999), 1247-1263.

W. R. Utz, A note on the Scholz-Brauer problem in addition chains. Proc. Amer. Math. Soc. 4, (1953). 462-463.

Vegh, Emanuel, A note on addition chains. J. Combinatorial Theory Ser. A 19 (1975), 117-118.

Whyburn, C. T., A note on addition chains. Proc. Amer. Math. Soc. 16 1965 1134.

LINKS

D. W. Wilson, Table of n, a(n) for n = 1..10001

Daniel Bleichenbacher, Efficiency and Security of Cryptosystems based on Number Theory. PhD Thesis, Diss. ETH No. 11404, Zuerich 1996. See p. 61.

Achim Flammenkamp, Shortest addition chains

D. E. Knuth, See the achain-all program

Alec Mihailovs, Notes on using Flammenkamp's tables

Hugo Pfoertner, Addition chains

Kari Ragnarsson, Bridget Eileen Tenner, Obtainable Sizes of Topologies on Finite Sets, Oct 06 2008, Journal of Combinatorial Theory, Series A 117 (2010) 138-151.

Eric Weisstein's World of Mathematics, Addition Chain

Eric Weisstein's World of Mathematics, Scholz Conjecture

Index to sequences related to the complexity of n

FORMULA

It seems that, for n>1, ln(n) < a(n) < (5/2)*ln(n); lim n ->infinity sum(k=1, n, a(k))/(n*ln(n)-n) = C = 1.8(4).... - Benoit Cloitre Oct 30 2002

a(n*m) <= a(n)+a(m). In particular, a(n^k) <= k * a(n). - Max Alekseyev, Jul 22, 2005

For all n => 2, a(n) =< (4/3)floor(log_2 n) + 2. [Jonathan Vos Post, Oct 08 2008]

EXAMPLE

For n < 149 and for many higher values of n, a(n) is the depth of n in a tree whose first 6 levels are shown below. The path from the root of the tree to n gives an optimal addition chain. (See Knuth, Vol. 2, Sect. 4.6.3, Fig. 14 and Ex. 5):

................1

................|

................2

.............../..\

............../.....\

............3..........4

........./....\.........\

......../......\.........\

.....5..........6..........8

..../..\........|......../...\

..7....10......12......9.......16

./..../..\..../..\..../.\...../..\

14...11..20..15..24..13..17..18..32

E.g., a(15) = 5 and an optimal chain for 15 is 1, 2, 3, 6, 12, 15

It is not possible to extend the tree to include the optimal addition chains for all n. For example, the chains for 43, 77, and 149 are incompatible. See the link to Achim Flammenkamp's web page on addition chains.

CROSSREFS

Cf. A003064, A003065, A005766, A230528

Sequence in context: A122953 A128998 A137813 * A117497 A117498 A064097

Adjacent sequences:  A003310 A003311 A003312 * A003314 A003315 A003316

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Jud McCranie, Nov 01 2001

Replaced arxiv URL by non-cached version - R. J. Mathar, Oct 07 2009

STATUS

approved

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Last modified November 24 01:09 EST 2014. Contains 249867 sequences.