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

1,3

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).

Example

101000_10 = 1100010{101000}1000_2.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A136809 A136813 A225237 * A181891 A115846 A352103

Adjacent sequences: A038099 A038100 A038101 * A038103 A038104 A038105

Keyword

nonn,base,easy

Author

Patrick De Geest, Feb 15 1999

Status

approved