cp's OEIS Frontend

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

%I A079599 #21 Feb 18 2015 07:51:57
%S A079599 0,2,8,10,16,18,24,26,32,34,40,42,48,50,56,58,64,66,72,74,80,82,88,90,
%T A079599 96,98,104,106,112,114,120,122,128,130,136,138,144,146,152,154,160,
%U A079599 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
%N A079599 Numbers n for which the n-th impartial game is a second player win.
%C A079599 These are the indices n for which A034798(n) = 0.
%C A079599 From _Antti Karttunen_, Jan 30 2014: (Start)
%C A079599 A236678(a(n)) = n+1 for all n.
%C A079599 Differs from A047467 for the first time at a(64).
%C A079599 Differs from A126002(n+1) for the first time not later than at n=281474976710656 (= 2^48), as:
%C A079599 a((2^48)-1) = a(281474976710655) = 18085043209519168250 < 18446744073709551616 (= 2^64), while
%C A079599 a(2^48) = a(281474976710656) = 36893488147419103232 > 2^64.
%C A079599 (End)
%D A079599 J. H. Conway, On numbers and games.
%H A079599 Antti Karttunen, <a href="/A079599/b079599.txt">Table of n, a(n) for n = 0..16383</a>
%H A079599 <a href="/index/Se#sequences_which_agree_for_a_long_time">Index entries for sequences which agree for a long time but are different</a>
%F A079599 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
%F A079599 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
%e A079599 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).
%o A079599 (Scheme)
%o A079599 (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)))))
%Y A079599 Characteristic function: A236677, its partial sums: A236678.
%Y A079599 Cf. A034798, A034797, A047467, A126002.
%K A079599 nonn
%O A079599 0,2
%A A079599 _Rob Arthan_, Jan 28 2003
%E A079599 More terms from _Antti Karttunen_, Jan 29 2014