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 A320441 Numbers whose binary expansion is quasiperiodic. 1

%I

%S 3,7,10,15,21,31,36,42,45,54,63,73,85,91,109,127,136,146,153,170,173,

%T 181,182,187,204,219,221,238,255,273,292,307,341,365,375,409,438,443,

%U 477,511,528,546,561,585,594,614,627,660,682,685,693,725,726,731,750

%N Numbers whose binary expansion is quasiperiodic.

%C The binary representation of a term (ignoring leading zeros) can be covered by (possibly overlapping) occurrences of one of its proper prefix.

%C This sequence contains A121016.

%C For any k > 0, there are A320434(k)/2 terms with binary length k.

%H Rémy Sigrist, <a href="/A320441/a320441.png">Scatterplot of the first difference of the first 100000 terms</a>

%F A020330(a(n)) belongs to the sequence for any n > 0.

%F A297405(a(n)) belongs to the sequence for any n > 0.

%e The first terms, alongside their binary representations and prefixes, are:

%e n a(n) bin(a(n)) prefix

%e -- ---- --------- ------

%e 1 3 11 1

%e 2 7 111 1

%e 3 10 1010 10

%e 4 15 1111 1

%e 5 21 10101 101

%e 6 31 11111 1

%e 7 36 100100 100

%e 8 42 101010 10

%e 9 45 101101 101

%e 10 54 110110 110

%e 11 63 111111 1

%e 12 73 1001001 1001

%o (PARI) isok(w) = { my (tt=0); for (l=1, oo, my (t=w%(2^l)); if (t!=tt, if (t==w, return (0)); my (r=w, g=l); while (g-->=0 && r>=t, r \= 2; if (r%

%o (2^l)==t, if (r==t, return (1), g=l))); tt = t)) }

%Y Cf. A020330, A121016, A297405, A320434.

%K nonn,base

%O 1,1

%A _Rémy Sigrist_, Jan 09 2019

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Last modified February 25 22:55 EST 2020. Contains 332270 sequences. (Running on oeis4.)