A005766 -id:A005766 - cp's OEIS frontend

A003313 Length of shortest addition chain for n.

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

Offset: 1

N. J. A. Sloane

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

			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.

Hatem M. Bahig, Mohamed H. El-Zahar, and Ken Nakamula, 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.
D. Bleichenbacher and A. Flammenkamp, An Efficient Algorithm for Computing Shortest Addition Chains, Preprint, 1997.
A. Flammenkamp, Drei Beitraege zur diskreten Mathematik: Additionsketten, No-Three-in-Line-Problem, Sociable Numbers, Diplomarbeit, Bielefeld 1991.
S. B. Gashkov and V. V. Kochergin, 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.
A. A. Gioia and M. V. Subbarao, 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.
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.
Michael O. Rabin and Shmuel Winograd, "Fast evaluation of polynomials by rational preparation." Communications on Pure and Applied Mathematics 25.4 (1972): 433-458. See Table p. 455.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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, and C. Duboc, Addition chains using continued fractions, J. Algorithms 10 (1989), 403-412.
Wawrzyniec Bieniawski, Piotr Masierak, Andrzej Tomski, and Szymon Łukaszyk, Assembly Theory - Formalizing Assembly Spaces and Discovering Patterns and Bounds, Preprints.org (2025).
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). 736-739.
John M. Campbell, A binary version of the Mahler-Popken complexity function, arXiv:2403.20073 [math.NT], 2024. See pp. 3-4. See also Integers (2024) Vol. 24, Art. No. A94. See pp. 3, 10.
Peter Downey, Benton Leong, and Ravi Sethi, Computing sequences with addition chains SIAM J. Comput. 10 (1981), 638-646.
M. Elia and F. Neri, A note on addition chains and some related conjectures, (Naples/Positano, 1988), pp. 166-181 of R. M. Capocelli, ed., Sequences, Springer-Verlag, NY 1990.
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 77-81.
Achim Flammenkamp, Shortest addition chains
A. A. Gioia, M. V. Subbarao, and M. Sugunamma, The Scholz-Brauer problem in addition chains, Duke Math. J. 29 1962 481-487.
Anastasiya Gorodilova, Sergey Agievich, Claude Carlet, Evgeny Gorkunov, Valeriya Idrisova, Nikolay Kolomeec, Alexandr Kutsenko, Svetla Nikova, Alexey Oblaukhov, Stjepan Picek, Bart Preneel, Vincent Rijmen, Natalia Tokareva, Problems and solutions of the Fourth International Students' Olympiad in Cryptography NSUCRYPTO, arXiv:1806.02059 [cs.CR], 2018.
R. L. Graham, A. C.-C. Yao, and F. F. Yao, Addition chains with multiplicative cost Discrete Math. 23 (1978), 115-119.
D. Knuth, Letter to N. J. A. Sloane, date unknown
D. E. Knuth, See the achain-all program
D. P. McCarthy, Effect of improved multiplication efficiency on exponentiation algorithms derived from addition chains Math. Comp. 46 (1986), 603-608.
Alec Mihailovs, Notes on using Flammenkamp's tables
Jorge Olivos, On vectorial addition chains J. Algorithms 2 (1981), 13-21.
Hugo Pfoertner, Addition chains
Kari Ragnarsson and Bridget Eileen Tenner, Obtainable Sizes of Topologies on Finite Sets, arXiv:0802.2550 [math.CO], 2008-2009; Journal of Combinatorial Theory, Series A 117 (2010) 138-151.
Arnold Schönhage, A lower bound for the length of addition chains Theor. Comput. Sci. 1 (1975), 1-12.
Edward G. Thurber, The Scholz-Brauer problem on addition chains Pacific J. Math. 49 (1973), 229-242.
Edward G. Thurber, On addition chains l(mn)<=l(n)-b and lower bounds for c(r) Duke Math. J. 40 (1973), 907-913.
Edward G. Thurber, Addition chains and solutions of l(2n)=l(n) and l(2^n-1)=n+l(n)-1, Discrete Math., Vol. 16 (1976), 279-289.
Edward G. Thurber, Addition chains-an erratic sequence Discrete Math. 122 (1993), 287-305.
Edward G. Thurber, 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.
Emanuel Vegh, A note on addition chains, J. Combinatorial Theory Ser. A 19 (1975), 117-118.
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

Cf. A003064, A003065, A005766, A014701, A079300, A230528.

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)
a(n) <= A014701(n). - Charles R Greathouse IV, Jan 03 2018
From Szymon Lukaszyk, Apr 05 2024: (Start)
For n = 2^s, a(n)=s;
for n = 2^s + 2^m, m in [0..s-1] (A048645), a(n)=s+1;
for n = 2^s + 3*2^m, m in [0..s-2] (A072823), a(n)=s+2;
for n = 2^s + 7*2^(s-3), s>2 (A072823), a(n)=s+2.(End)

More terms from Jud McCranie, Nov 01 2001

A089265 a(1) = 0; thereafter a(2n) = a(n) + 1, a(2n+1) = 2*n.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 6, 3, 8, 5, 10, 4, 12, 7, 14, 4, 16, 9, 18, 6, 20, 11, 22, 5, 24, 13, 26, 8, 28, 15, 30, 5, 32, 17, 34, 10, 36, 19, 38, 7, 40, 21, 42, 12, 44, 23, 46, 6, 48, 25, 50, 14, 52, 27, 54, 9, 56, 29, 58, 16, 60, 31, 62, 6, 64, 33, 66, 18, 68, 35, 70, 11, 72

Offset: 1

Ralf Stephan, Oct 30 2003

In the binary representation of n, swallow all zeros from the right, then add the number of swallowed zeros, and subtract 1. - Ralf Stephan, Aug 22 2013

First differences of A005766.

Cf. A003602, A220466.

nmax:=73: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := p  + 2*(n-1) od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Jan 23 2013

a[n_] := With[{v = IntegerExponent[n, 2]}, v + n/2^v - 1];
Array[a, 100] (* Jean-François Alcover, Feb 28 2019 *)

a(n) = valuation(n,2) + n/2^valuation(n,2) - 1

a(1) = 0; thereafter a(2*n) = a(n) + 1, a(2*n+1) = 2*n.
a(n) = A007814(n) + 2*A025480(n-1) = A007814(n) + A000265(n) - 1.
G.f.: sum(k>=0, (t^2+2t^3-t^4)/(1-t^2)^2, t=(x^2)^k).
a((2*n-1)*2^p) = p + 2*(n-1), p >= 0. - Johannes W. Meijer, Jan 23 2013

A008057 Smallest sum of an addition chain for 2n+1.

Original entry on oeis.org

0, 5, 10, 16, 20, 27, 31, 35, 40, 47, 51, 56, 60, 65, 74, 78, 80, 86, 92, 96, 102, 106, 110, 120, 121, 125, 134, 137, 142, 148, 153, 156, 160, 167, 171, 182, 184, 185, 192, 201, 200, 206, 210, 219, 227, 231, 233, 237, 241, 245

Offset: 0

N. J. A. Sloane, Aug 07 2003

			The smallest chain for 5 is 2, 3, 5 with sum a(2) = 2+3+5 = 10.
The smallest chain for 7 is 2, 3, 4, 7 with sum a(3) = 2+3+4+7 = 16.

H. Zantema, Minimizing sums of addition chains, RUU-CS-89-15 (1989).
H. Zantema, Minimizing sums of addition chains, J. Algorithms 12 (1991) 281-307.
Index to sequences related to the complexity of n

Cf. A003313, A079300, A003065, A005766, A376449.

step(V)=my(U=List(),v); for(i=1,#V, v=V[i]; for(i=1,#v, for(j=i,#v, if(v[i]+v[j]>v[#v], listput(U, concat(v, v[i]+v[j])))))); vecsort(Vec(U),,8)
sm(v)=sum(i=2,#v,v[i])
a(n)=if(n<2,return(5*n)); n=2*n+1; my(V=[[1,2]],U,t); while(#(U=select(v->v[#v]==n,V))==0, V=select(v->v[#v]<=n,step(V))); t=vecmin(apply(sm,U)); while(#V, V=step(select(v->sm(v)Charles R Greathouse IV, Jul 17 2013

a(n) = A376449(2*n+1) + 2*n. - Pontus von Brömssen, Apr 22 2025

a(30)-a(46) from Sean A. Irvine, Mar 08 2018

a(47)-a(49) (using A376449 b-file) from Pontus von Brömssen, Apr 22 2025

A086833 Minimum number of different addends occurring in any shortest addition chain of Brauer type for a given n, or 0 if n has no shortest addition chain of Brauer type.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 4, 3, 4, 5, 4, 4, 4, 3, 4, 4, 4, 4, 5, 5, 5, 4, 4, 4, 4, 4, 5, 5, 4, 6, 5, 4, 6, 4, 5, 5, 5, 5, 5, 5, 4, 4, 5, 4, 5, 5, 5, 5, 5, 4, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 7, 5, 5, 6, 4, 6, 7, 5, 6, 7, 5, 6, 6, 5, 5, 7, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5

Offset: 1

Tatsuru Murai, Aug 08 2003

nonn

n = 12509 is the first n for which a(n) = 0 because it is the smallest number that has no shortest addition chain of Brauer type. - Hugo Pfoertner, Jun 10 2006 [Edited by Pontus von Brömssen, Apr 25 2025]

			a(23)=5 because 23=1+1+2+1+4+9+5 is the shortest addition chain for 23.
For n=9 there are A079301(9)=3 different shortest addition chains, all of Brauer type:
[1 2 3 6 9] -> 9=1+1+1+3+3 -> 2 different addends {1,3}
[1 2 4 5 9] -> 9=1+1+2+1+4 -> 3 different addends {1,2,4}
[1 2 4 8 9] -> 9=1+1+2+4+1 -> 3 different addends {1,2,4}
The minimum number of different addends is 2, therefore a(9)=2.

Giovanni Resta, Tables of Shortest Addition Chains, computed by David W. Wilson.
Index to sequences related to the complexity of n

Cf. A003064, A003065, A003313, A005766, A008057, A008928, A008933, A079300, A079300, A079301, A349044.

a(n) = 0 if and only if n is in A349044. - Pontus von Brömssen, Apr 25 2025

Edited by Hugo Pfoertner, Jun 10 2006

Escape clause added by Pontus von Brömssen, Apr 25 2025

cp's OEIS Frontend

A003313 Length of shortest addition chain for n.

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A089265 a(1) = 0; thereafter a(2*n) = a(n) + 1, a(2*n+1) = 2*n.

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A008057 Smallest sum of an addition chain for 2n+1.

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A086833 Minimum number of different addends occurring in any shortest addition chain of Brauer type for a given n, or 0 if n has no shortest addition chain of Brauer type.

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Examples

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Crossrefs

Formula

Extensions

A089265 a(1) = 0; thereafter a(2n) = a(n) + 1, a(2n+1) = 2*n.