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

Author

Patrick De Geest, Feb 15 1999

Keywords

• nonn
• base
• easy

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).

Crossrefs

Cf. A038103, A038104, A038105, A038106, A228050, A228051, A228052, A227549, A350572.

Programs

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