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A113945
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Numbers n such that the smallest possible number of multiplications required to compute x^n is by 1 less than the number of multiplications obtained by Knuth's power tree method.
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5
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77, 154, 233, 293, 308, 319, 359, 367, 377, 382, 423, 457, 466, 551, 553, 559, 571, 573, 586, 616, 617, 619, 623, 638, 699, 713, 717, 718, 734, 754, 764, 813, 841, 846, 849, 869, 879, 905, 914, 932, 1007, 1051, 1063, 1069, 1102, 1103, 1106, 1115, 1118, 1133
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OFFSET
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1,1
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COMMENTS
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The first three terms are given in Knuth's TAOCP, Vol. 2. The sequence is based on a table of shortest addition chain lengths computed by Neill M. Clift, see link to Achim Flammenkamp's web page given at A003313.
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REFERENCES
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D. E. Knuth, The Art of Computer Programming Third Edition. Vol. 2, Seminumerical Algorithms. Chapter 4.6.3 Evaluation of Powers, Page 464. Addison-Wesley, Reading, MA, 1997.
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LINKS
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EXAMPLE
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a(1)=77 because the power tree construction produces the chain 1 2 3 5 7 14 19 38 76 77 requiring 9 additions, whereas there are 4 shortest chains that come along with 8 additions, e.g. 1 2 4 8 9 17 34 43 77.
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CROSSREFS
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Cf. A114622 [The power tree (as defined by Knuth)], A003313 [Length of shortest addition chain for n], A115614 [numbers such that Knuth's power tree method produces a result deficient by 2], A115615 [numbers such that Knuth's power tree method produces a result deficient by 3], A115616 [smallest number for which Knuth's power tree method produces an addition chain n terms longer than the shortest possible chain].
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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