cp's OEIS Frontend

A225238 Numbers n such that n occurs within its base 2 representation regarded as a fixed necklace, but n is not a substring of the base 2 representation regarded as a string.

%I A225238 #14 Feb 16 2025 08:33:19
%S A225238 1111,110110,111011,1011110,1101111,10011001,10100110,11001010,
%T A225238 11010011,110011111,111010011,111011101,1001101111,11001101111,
%U A225238 11010010100,100101110100,101000111011,101011001010,101111110111,110011001100,110111001101,111011110011
%N A225238 Numbers n such that n occurs within its base 2 representation regarded as a fixed necklace, but n is not a substring of the base 2 representation regarded as a string.
%C A225238 No power of 10 can occur in this sequence.
%H A225238 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/Necklace.html">MathWorld: Necklace</a>
%e A225238 111011 (in base 10) = 11011000110100011 (in base 2). Regarding this base 2 representation as a fixed necklace, we can list characters in the order 11110110001101000 by starting with the characters "11" at the end of the base 2 representation. In this listing 111011 occurs (1{111011}0001101000). 111011 however does not occur in the original base 2 representation 11011000110100011. Thus 111011 is in the sequence.
%o A225238 (PARI) {inseq(w)=local(bw,mm,texp,btod,bigb,lbb,swsq,ii, hwf);
%o A225238 bw=binary(w); mm=length(bw); texp=0; btod=0;
%o A225238 forstep(i=mm, 1, -1, btod=btod+bw[i]*10^texp; texp++);
%o A225238 bigb=binary(btod); lbb=length(bigb);
%o A225238 for(k=0, lbb - 1 , swsq=1;
%o A225238 for(j=1, mm, ii=(j+k)%lbb; if(ii==0, ii=lbb);
%o A225238 if(bw[j]!=bigb[ii], swsq=-1)); if(swsq==1, hwf=k; break));
%o A225238 if(swsq==1,if(hwf>lbb-mm, swsq=btod, swsq=-1)); return(swsq)}
%o A225238 {ptd=0;for(w=0, 10^9, jj=inseq(w); if(jj>=0, ptd++; print1(jj,", "); if(ptd>23,break)))}
%Y A225238 The union of A038102 and this sequence is A225237.
%K A225238 nonn,base
%O A225238 1,1
%A A225238 _Douglas Latimer_, May 04 2013