A115016 a(n) is the smallest number k that has a shortest addition chain whose length A003313(k) = A003313(n*k), or 0 if this never happens.

1, 191, 171, 30958077, 3277, 2731, 28087

Offset: 1

Author

Hugo Pfoertner, Feb 26 2006

Keywords

• nonn
• hard
• more

Comments

Using ? to indicate a term whose value is presently unknown, the sequence reads 1, 191, 171, 30958077, 3277, 2731, 28087, ?, 233017, 432541, 953251, 699051, 12905551, 1797559, ?, ?, ?, ?, 7064091, ... This is based on several years work using a variety of algorithms. - Neill M. Clift, May 23 2008

It was conjectured that no shortest addition chains exist such that A003313(m)=A003313(m*2^k) for k>1. This is now known to be false, since a(4) != 0.

Examples

			a(3)=171 because 171 and 513=3*171 both have a shortest addition chain of length 10. 171 and 513 is the smallest pair of numbers with the property A003313(k)=A003313(3*k). Examples for the corresponding shortest chains are [1 2 4 5 7 14 19 38 57 114 171] and [1 2 4 8 16 32 64 128 256 512 513].
		

References

• For a comprehensive list of references see A003313.
• See also D. E. Knuth, updates to Vol. 2 of TAOCP.

Links

• Daniel Bleichenbacher, Efficiency and Security of Cryptosystems based on Number Theory, PhD Thesis, Diss. ETH No. 11404, Zürich 1996; p. 61.
• Neill Michael Clift, Calculating optimal addition chains, Computing 91.3 (2011): 265-284.

Crossrefs

Cf. A003313 [l(k)], A086878 [l(k)=l(2*k)], A116459 [l(k)=l(3*k)], A261986 [l(k)=l(4*k)], A116460 [l(k)=l(5*k)], A116461 [l(k)=l(6*k)], A116462 [l(k)=l(7*k)], A116463 [l(k)=l(9*k)], A117151 [l(k)=l(10*k)].

Extensions

a(4) from Neill M. Clift, May 21 2008