A079599 Numbers n for which the n-th impartial game is a second player win.

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

Offset: 0

Author

Rob Arthan, Jan 28 2003

Keywords

• nonn

Comments

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)

Examples

			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).
		

References

• J. H. Conway, On numbers and games.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..16383
• Index entries for sequences which agree for a long time but are different

Crossrefs

Characteristic function: A236677, its partial sums: A236678.

Cf. A034798, A034797, A047467, A126002.

Programs

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)))))

Formula

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

Extensions

More terms from Antti Karttunen, Jan 29 2014