A161579 Positions n such that A010060(n) = A010060(n+3).

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, 38, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 54, 56, 57, 59, 60, 61, 63, 64, 65, 67, 68, 70, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 86, 88, 89, 91, 92, 94, 96, 97, 99, 100, 102, 104, 105, 107

Offset: 1

Author

Vladimir Shevelev, Jun 14 2009

Keywords

• nonn
• base

Comments

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.

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, arXiv:1401.3727 [math.NT], 2014.
• J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, J. de Théorie des Nombres de Bordeaux, 27, no. 2 (2015), 375-388.
• V. Shevelev, Equations of the form t(x+a)=t(x) and t(x+a)=1-t(x) for Thue-Morse sequence arXiv:0907.0880 [math.NT], 2009-2012. [_Vladimir Shevelev_, Jul 31 2009]

Crossrefs

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

Programs

Mathematica

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 *)

PARI

is(n)=hammingweight(n)%2==hammingweight(n+3)%2 \\ Charles R Greathouse IV, Aug 20 2013

Formula

Equals {A001477} \ {A161580}.

Extensions

More terms from R. J. Mathar, Aug 17 2009