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.

1111, 110110, 111011, 1011110, 1101111, 10011001, 10100110, 11001010, 11010011, 110011111, 111010011, 111011101, 1001101111, 11001101111, 11010010100, 100101110100, 101000111011, 101011001010, 101111110111, 110011001100, 110111001101, 111011110011

Offset: 1

Author

Douglas Latimer, May 04 2013

Keywords

• nonn
• base

Comments

No power of 10 can occur in this sequence.

Examples

			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.
		

Links

• Eric W. Weisstein, MathWorld: Necklace

Crossrefs

The union of A038102 and this sequence is A225237.

Programs

PARI

{inseq(w)=local(bw,mm,texp,btod,bigb,lbb,swsq,ii, hwf);
bw=binary(w); mm=length(bw); texp=0; btod=0;
forstep(i=mm, 1, -1, btod=btod+bw[i]*10^texp; texp++);
bigb=binary(btod); lbb=length(bigb);
for(k=0, lbb - 1 , swsq=1;
for(j=1, mm, ii=(j+k)%lbb; if(ii==0, ii=lbb);
if(bw[j]!=bigb[ii], swsq=-1)); if(swsq==1, hwf=k; break));
if(swsq==1,if(hwf>lbb-mm, swsq=btod, swsq=-1)); return(swsq)}
{ptd=0;for(w=0, 10^9, jj=inseq(w); if(jj>=0, ptd++; print1(jj,", "); if(ptd>23,break)))}