A038102 Numbers k such that k is a substring of its base-2 representation.
0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1100, 1101, 10000, 10001, 10011, 10100, 10101, 10111, 11000, 11001, 11100, 11101, 100000, 100001, 101000, 101010, 101100, 101101, 101111, 110000, 110001, 110101, 111100, 111101, 1000000
Offset: 1
Examples
101000_10 = 1100010{101000}1000_2.
Links
- Hans Havermann, Table of n, a(n) for n = 1..1512 (terms 1..1000 from Giovanni Resta, terms 1001..1167 from Robert G. Wilson v).
- Hans Havermann, pdf file showing the corresponding A350572 terms and illustrating the binary embeddings (for terms 1..1512).
Programs
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Mathematica
Select[FromDigits /@ IntegerDigits[Range[2^15]-1, 2], StringPosition[StringJoin @@ (ToString /@ IntegerDigits[#, 2]), ToString@#] != {} &] (* terms < 10^15, Giovanni Resta, Apr 30 2013 *) f[n_] := Block[{a = FromDigits@ IntegerDigits[n, 2]}, If[ StringPosition[ ToString@ FromDigits@ IntegerDigits[ a, 2], ToString@ a] != {}, a, 0]]; k = 0; lst = {}; While[k < 65, AppendTo[lst, f@k]; lst = Union@ lst; k++]; lst (* Robert G. Wilson v, Jun 29 2014 *) -
PARI
{for(vv=0, 200, bvv=binary(vv); mm=length(bvv); texp=0; btod=0; forstep(i=mm, 1, -1, btod=btod+bvv[i]*10^texp; texp++); bigb=binary(btod); lbb=length(bigb); swsq=1; for(k=0, lbb - mm , for(j=1, mm, if(bvv[j]!=bigb[j+k], swsq=0)); if(swsq==1, print1(btod, ", "); break, swsq=1)))} \\\ Douglas Latimer, Apr 29 2013 -
Python
from itertools import count, islice, product def ok(n): return int(max(str(n))) < 2 and str(n) in bin(n) def agen(): # generator of terms yield 0 for d in count(1): for rest in product("01", repeat=d-1): k = int("1" + "".join(rest)) if ok(k): yield k print(list(islice(agen(), 35))) # Michael S. Branicky, Jan 04 2022
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