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 A128998 Length of shortest addition-subtraction chain for n. 2
 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, 6, 5, 6, 6, 7, 6, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 8, 7, 8, 7, 8, 8, 8, 7, 8, 7, 7, 6, 7, 7, 8, 7, 8, 8, 8, 7, 8, 8, 8, 8, 8, 8, 8, 7, 8, 8, 8, 8, 8, 8, 9 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Equivalently, the minimal total number of multiplications and divisions required to compute an n-th power. This is useful for exponentiation on, for example, elliptic curves where division is cheap (as proposed by Morain and Olivos, 1990). Addition-subtraction chains are also defined for negative n. Various bounds and a rules to construct a(n) up to n=42 can be found in Volger (1985). a(n) < A003313(n) for n in A229624. - T. D. Noe, May 02 2007 REFERENCES Hugo Volger, Some results on addition/subtraction chains, Information Processing Letters, Vol. 20 (1985), pp. 155-160. LINKS F. Morain and J. Olivos, Speeding up the computations on an elliptic curve using addition-subtraction chains, RAIRO Informatique theoretique et application, vol. 24 (1990), pp. 531-543. EXAMPLE For example, a(31) = 6 because 31 = 2^5 - 1 and 2^5 can be produced by 5 additions (5 doublings) starting with 1. CROSSREFS Cf. A003313. Sequence in context: A122953 A259847 A259103 * A137813 A003313 A277608 Adjacent sequences:  A128995 A128996 A128997 * A128999 A129000 A129001 KEYWORD nonn,nice AUTHOR Steven G. Johnson (stevenj(AT)math.mit.edu), May 01 2007 EXTENSIONS More terms from T. D. Noe, May 02 2007 STATUS approved

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Last modified March 19 17:05 EDT 2019. Contains 321330 sequences. (Running on oeis4.)