A003065 Number of integers with a shortest addition chain of length n.

1, 1, 2, 3, 5, 9, 15, 26, 44, 78, 136, 246, 432, 772, 1382, 2481, 4490, 8170, 14866, 27128, 49544, 90371, 165432, 303475, 558275, 1028508, 1896704, 3501029, 6465774, 11947258, 22087489, 40886910, 75763102, 140588339, 261070184, 485074788, 901751654, 1677060520

Offset: 0

Author

N. J. A. Sloane, Don Knuth

Keywords

• nonn
• nice
• hard

Examples

			a(6) = 15 because 15 numbers have shortest addition chains involving 6 additions. These numbers are 19,21,22,23,25,26,27,28,30,33,34,36,40,48,64.
		

References

• D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 459.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• See A003313 for a much more extensive list of references and links.

Links

• Glen Whitney, Table of n, a(n) for n = 0..44 (Terms 0..38 from R. J. Mathar)
• Daniel Bleichenbacher, Efficiency and Security of Cryptosystems based on Number Theory, PhD Thesis, Diss. ETH No. 11404, Zürich 1996. See p. 61.
• M. Elia and F. Neri, A note on addition chains and some related conjectures, pp. 166-181 of R. M. Capocelli, ed., Sequences, Springer-Verlag, NY 1990.
• Achim Flammenkamp, Shortest addition chains
• Index to sequences related to the complexity of n

Crossrefs

Cf. A003064, A003313.

Cf. A114623 [Number of integers for which Knuth's power tree method produces an addition chain of length n].

Extensions

Updated through a(28) from Achim Flammenkamp's web site Feb 01 2005

a(28) corrected from 6465773 to 6465774, based on information received from Neill M. Clift. - Hugo Pfoertner, Jan 29 2006

a(29)-a(30) from Neill M. Clift, Jun 15 2007

a(31)-a(33) from Neill M. Clift, May 21 2008

b-file up to a(38) extracted from Achim Flammenkamp's web site. R. J. Mathar, May 14 2013

b-file updated to a(44) from Achim Flammenkamp's web site. Glen Whitney, Nov 09 2021