

A003313


Length of shortest addition chain for n.
(Formerly M0255)


52



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
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OFFSET

1,3


COMMENTS

Equivalently, minimal number of multiplications required to compute the nth power.


REFERENCES

Bahig, Hatem M.; ElZahar, Mohamed H.; Nakamula, Ken; Some results for some conjectures in addition chains, in Combinatorics, computability and logic, pp. 4754, Springer Ser. Discrete Math. Theor. Comput. Sci., Springer, London, 2001.
D. Bleichenbacher and A. Flammenkamp, An Efficient Algorithm for Computing Shortest Addition Chains, Preprint, 1997.
Elia, M. and Neri, F.; A note on addition chains and some related conjectures, in Sequences (Naples/Positano, 1988), pp. 166181, Springer, New York, 1990.
A. Flammenkamp, Drei Beitraege zur diskreten Mathematik: Additionsketten, NoThreeinLineProblem, 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), 2240, 119120; 1265027], Siberian Adv. Math. 4 (1994), 116.
Gioia, A. A. and Subbarao, M. V., The ScholzBrauer problem in addition chains, II, in Proceedings of the Eighth Manitoba Conference on Numerical Mathematics and Computing (Univ. Manitoba, Winnipeg, Man., 1978), pp. 251274, Congress. Numer., XXII, Utilitas Math., Winnipeg, Man., 1979.
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.
Rabin, Michael O., and Shmuel Winograd. "Fast evaluation of polynomials by rational preparation." Communications on Pure and Applied Mathematics25.4 (1972): 433458. See Table p. 455.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

D. W. Wilson and Antoine Mathys, Table of n, a(n) for n = 1..100000 (10001 terms from D. W. Wilson)
F. Bergeron, J. Berstel, S. Brlek, C Duboc, Addition chains using continued fractions, J. Algorithms 10 (1989), 403412.
Daniel Bleichenbacher, Efficiency and Security of Cryptosystems based on Number Theory. PhD Thesis, Diss. ETH No. 11404, Zürich 1996. See p. 61.
Alfred Brauer, On addition chains Bull. Amer. Math. Soc. 45, (1939). 736739.
Peter Downey, Benton Leong, and Ravi Sethi, Computing sequences with addition chains SIAM J. Comput. 10 (1981), 638646.
Christian Elsholtz et al, Upper bound on length of addition chain, Math Overflow, Sep 18 2015.
P. Erdős, Remarks on number theory. III. On addition chains, Acta Arith. 6 1960 7781.
Achim Flammenkamp, Shortest addition chains
A. A. Gioia, M. V. Subbarao, and M. Sugunamma, The ScholzBrauer problem in addition chains, Duke Math. J. 29 1962 481487.
R. L. Graham, A. C.C. Yao, F. F. Yao, Addition chains with multiplicative cost Discrete Math. 23 (1978), 115119.
D. E. Knuth, See the achainall program
D. P. McCarthy, Effect of improved multiplication efficiency on exponentiation algorithms derived from addition chains Math. Comp. 46 (1986), 603608.
Alec Mihailovs, Notes on using Flammenkamp's tables
Jorge Olivos, On vectorial addition chains J. Algorithms 2 (1981), 1321.
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) 138151.
Arnold Schönhage, A lower bound for the length of addition chains Theor. Comput. Sci. 1 (1975), 112.
Edward G. Thurber, The ScholzBrauer problem on addition chains Pacific J. Math. 49 (1973), 229242.
Edward G. Thurber, On addition chains l(mn)<=l(n)b and lower bounds for c(r) Duke Math. J. 40 (1973), 907913.
Edward G. Thurber, Addition chains and solutions of l(2n)=l(n) and l(2^n1)=n+l(n)1, Discrete Math., Vol. 16 (1976), 279289.
Edward G. Thurber, Addition chainsan erratic sequence Discrete Math. 122 (1993), 287305.
Edward G. Thurber, Efficient generation of minimal length addition chains, SIAM J. Comput. 28 (1999), 12471263.
W. R. Utz, A note on the ScholzBrauer problem in addition chains, Proc. Amer. Math. Soc. 4, (1953). 462463.
Emanuel Vegh, A note on addition chains, J. Combinatorial Theory Ser. A 19 (1975), 117118.
Eric Weisstein's World of Mathematics, Addition Chain
Eric Weisstein's World of Mathematics, Scholz Conjecture
C. T. Whyburn, A note on addition chains Proc. Amer. Math. Soc. 16 1965 1134.
Index to sequences related to the complexity of n


FORMULA

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
From Achim Flammenkamp, Oct 26 2016: (Start)
a(n) <= 9/log_2(71) log_2(n), for all n.
It is conjectured by D. E. Knuth, K. Stolarsky et al. that for all n: floor(log_2(n)) + ceiling(log_2(v(n))) <= a(n). (End)


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: A259103 A128998 A137813 * A277608 A117497 A117498
Adjacent sequences: A003310 A003311 A003312 * A003314 A003315 A003316


KEYWORD

nonn,nice,look,changed


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Jud McCranie, Nov 01 2001
arXiv URL replaced with noncached version by R. J. Mathar, Oct 07 2009


STATUS

approved



