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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

Table of n, a(n) for n=1..87.

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.

Index to sequences related to the complexity of n

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: A208280 A139141 A122953 * A137813 A003313 A117497

Adjacent sequences:  A128995 A128996 A128997 * A128999 A129000 A129001

KEYWORD

nonn,nice,changed

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 October 24 11:46 EDT 2014. Contains 248516 sequences.