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A079599
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Numbers n for which the n-th impartial game is a second player win.
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6
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0, 2, 8, 10, 16, 18, 24, 26, 32, 34, 40, 42, 48, 50, 56, 58, 64, 66, 72, 74, 80, 82, 88, 90, 96, 98, 104, 106, 112, 114, 120, 122, 128, 130, 136, 138, 144, 146, 152, 154, 160, 162, 168, 170, 176, 178, 184, 186, 192, 194, 200, 202, 208, 210, 216, 218, 224, 226, 232, 234, 240, 242, 248, 250, 512, 514
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OFFSET
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0,2
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COMMENTS
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These are the indices n for which A034798(n) = 0.
From Antti Karttunen, Jan 30 2014: (Start)
A236678(a(n)) = n+1 for all n.
Differs from A047467 for the first time at a(64).
Differs from A126002(n+1) for the first time not later than at n=281474976710656 (= 2^48), as:
a((2^48)-1) = a(281474976710655) = 18085043209519168250 < 18446744073709551616 (= 2^64), while
a(2^48) = a(281474976710656) = 36893488147419103232 > 2^64.
(End)
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REFERENCES
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J. H. Conway, On numbers and games.
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences which agree for a long time but are different
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FORMULA
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a(0) = 0; a(n+1) = least x > a(n) such that the coefficient of 2^a(j) is zero in the binary expansion of x for all j < n+1
Alternatively: rewrite the binary representation of n in such a way that the forbidden bit-positions given by this sequence (with bit-position 0 standing for the least significant bit) are vacated, by shifting the rest of bits one bit left. E.g., bit-positions 0, 2, 8, 10, ... are forbidden, thus we rewrite 1 to 1x = 10 = 2, 2 (in binary 10) to 1x0x = 1000 = 8, 3 (in binary 11) to 1x1x = 1010 = 10, 4 (in binary 100) to 10x0x = 1000 = 16, 64 (in binary 1000000) to 1x00000x0x = 1000000000 = 512, etc. - Antti Karttunen, Jan 30 2014
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EXAMPLE
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a(1) = 2 (rather than 1) because 1 = 2^0 = 2^a(0); a(64) = 512 (rather than 256) because 256 = 2^8 = 2^a(2).
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PROG
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(Scheme)
(define (A079599 n) (let loop ((n n) (i 0) (j 0) (s 0)) (cond ((zero? n) s) ((odd? n) (loop (/ (- n 1) 2) (+ i 1) (+ j 1 (A236677 j)) (+ s (expt 2 (+ j (A236677 j)))))) (else (loop (/ n 2) (+ i 1) (+ j 1 (A236677 j)) s)))))
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CROSSREFS
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Characteristic function: A236677, its partial sums: A236678.
Cf. A034798, A034797, A047467, A126002.
Sequence in context: A073886 A079930 A047467 * A126002 A110913 A038638
Adjacent sequences: A079596 A079597 A079598 * A079600 A079601 A079602
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KEYWORD
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nonn
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AUTHOR
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Rob Arthan, Jan 28 2003
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EXTENSIONS
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More terms from Antti Karttunen, Jan 29 2014
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STATUS
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approved
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