A161579 - OEIS

%I #22 Apr 16 2018 02:52:17

%S 0,1,3,4,6,8,9,11,12,13,15,16,17,19,20,22,24,25,27,28,30,32,33,35,36,

%T 38,40,41,43,44,45,47,48,49,51,52,54,56,57,59,60,61,63,64,65,67,68,70,

%U 72,73,75,76,77,79,80,81,83,84,86,88,89,91,92,94,96,97,99,100,102,104,105,107

%N Positions n such that A010060(n) = A010060(n+3).

%C Or: union of A131323 with the sequence of terms of the form A131323(n)-2, and with the sequence of terms of the form A036554(n)-2.

%C Conjecture: In every sequence of numbers n such that A010060(n)=A010060(n+k), for fixed odd k, the odious (A000069) and evil (A001969) terms alternate. - _Vladimir Shevelev_, Jul 31 2009

%H G. C. Greubel, <a href="/A161579/b161579.txt">Table of n, a(n) for n = 1..10000</a>

%H J.-P. Allouche, <a href="http://arxiv.org/abs/1401.3727">Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence</a>, arXiv:1401.3727 [math.NT], 2014.

%H J.-P. Allouche, <a href="http://dx.doi.org/10.5802/jtnb.906">Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence</a>, J. de Théorie des Nombres de Bordeaux, 27, no. 2 (2015), 375-388.

%H V. Shevelev, <a href="http://arxiv.org/abs/0907.0880">Equations of the form t(x+a)=t(x) and t(x+a)=1-t(x) for Thue-Morse sequence</a> arXiv:0907.0880 [math.NT], 2009-2012. [_Vladimir Shevelev_, Jul 31 2009]

%F Equals {A001477} \ {A161580}.

%t tm[0] = 0; tm[n_?EvenQ] := tm[n] = tm[n/2]; tm[n_] := tm[n] = 1 - tm[(n-1)/2]; Reap[For[n = 0, n <= 200, n++, If[tm[n] == tm[n+3], Sow[n]]]][[2, 1]] (* _Jean-François Alcover_, Oct 24 2013 *)

%o (PARI) is(n)=hammingweight(n)%2==hammingweight(n+3)%2 \\ _Charles R Greathouse IV_, Aug 20 2013

%Y Cf. A131323, A036554, A010060, A121539, A079523, A081706, A161580.

%K nonn,base

%O 1,3

%A _Vladimir Shevelev_, Jun 14 2009

%E More terms from _R. J. Mathar_, Aug 17 2009