A038106 -id:A038106 - cp's OEIS frontend

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

Patrick De Geest, Feb 15 1999

			101000_10 = 1100010{101000}1000_2.

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).

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

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 *)

{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

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

A228050 The decimal representation of n is a substring of its base 7 representation.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 5624032642, 5624033055, 5624104634, 5624105050, 5624110136, 102616333034, 102620103253, 103055445560, 206154633166, 206154633200, 212216263215, 212220033434, 315315450515, 321340554340, 424436332033, 424440102253, 430461435550

Offset: 1

Giovanni Resta, Aug 06 2013

			2523016430113651303122433 = (53252301643011365130312243352)_7.

Giovanni Resta, Table of n, a(n) for n = 1..156 (terms < 10^25)

Cf. A038102-A038106, A228051, A228052, A227549.

A228052 The decimal representation of n is a substring of its base 9 representation.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 42212277303475883580, 42212277303475883581, 42212277303475883582, 42212277303475883583, 42212277303475883584, 42212277303475883585, 42212277303475883586, 42212277303475883587, 42212277303475883588, 1066338786883726756382

Offset: 1

Giovanni Resta, Aug 06 2013

			42212277303475883588 = (342212277303475883588)_9.

Giovanni Resta, Table of n, a(n) for n = 1..103 (terms < 10^50)

Cf. A038102-A038106, A228050, A228051, A227549.

A038103 Numbers k such that k is a substring of its base-3 representation.

Original entry on oeis.org

0, 1, 2, 10, 20, 21, 102, 110, 210, 211, 212, 220, 1011, 1112, 1121, 2022, 12101, 12102, 12112, 12122, 10121021, 10121022, 12222212, 102121110, 102121120, 200121022, 1001120220, 2011001102, 2012012221, 2100221021, 2102111111

Offset: 1

Patrick De Geest, Feb 15 1999

			12101 = base 10 -> 121{12101}2 = base 3.

Hans Havermann, Table of n, a(n) for n = 1..210 (terms 1..151 from Giovanni Resta).
Hans Havermann, pdf file showing the corresponding A350573 terms and illustrating the ternary embeddings (for terms 1..210).

Cf. A038102-A038106, A228050-A228052, A227549, A350573.

from sympy.ntheory.digits import digits
from itertools import count, islice, product
def agen(): # generator of terms
    yield 0
    for d in count(1):
        for first in "12":
            for rest in product("012", repeat=d-1):
                s = first + "".join(rest)
                if s in "".join(str(d) for d in digits(int(s), 3)[1:]):
                    yield int(s)
print(list(islice(agen(), 31))) # Michael S. Branicky, Jan 08 2022

from itertools import count, islice
from gmpy2 import digits
def A038103_gen(): return (int(s) for s in (digits(n,3) for n in count(0)) if s in digits(int(s),3))
A038103_list = list(islice(A038103_gen(),30)) # Chai Wah Wu, Jan 09 2022

A228051 The decimal representation of n is a substring of its base 8 representation.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 21355040, 21355041, 21355042, 21355043, 21355044, 21355045, 21355046, 21355047, 5406016340533523126, 5406016341275264235, 5406016341324744317, 5406016341324744320, 5406016341325061711, 5406016341325061712, 5406016342066514511

Offset: 1

Giovanni Resta, Aug 06 2013

			5406016340533523126 = (454060163405335231266)_8.

Giovanni Resta, Table of n, a(n) for n = 1..169 (terms < 10^35)

Cf. A038102-A038106, A228050, A228052, A227549.

A038105 Numbers n with property that n is a substring of its base 5 representation.

Original entry on oeis.org

0, 1, 2, 3, 4, 10000, 10001, 10002, 10003, 10004, 20000, 20001, 20002, 20003, 20004, 30000, 30001, 30002, 30003, 30004, 40000, 40001, 40002, 40003, 40004, 124343, 124401, 130000, 130001, 130002, 130003, 130004, 132114, 210000, 210001, 210002

Offset: 1

Patrick De Geest, Feb 15 1999

			124343 = base 10 -> {124343}33 = base 5.

Giovanni Resta, Table of n, a(n) for n = 1..445 (terms < 10^15)

Cf. A038102-A038106, A228050-A228052, A227549.

A307257 Numbers k such that k is the substring identical to the most significant digits of its base-6 representation.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 155133423353, 1535554025003332541, 1541322512150022100, 14544222532144144105400055, 15313352033401001032012412, 144510301032554451200342301123415, 145033105231511102401034455341005, 152142233320210504134120425450055, 1435322205115414041210434301130104431115

Offset: 1

Scott R. Shannon, Apr 01 2019

Numbers k whose base-6 representation begins with the same digits as k itself.

			155133423353_10 = 155133423353145_6, which also begins with '155133423353'.

Scott R. Shannon, Table of n, a(n) for n = 1..500

This is a subsequence of A038106.

isok(n) = my(vb=digits(n, 6), vd=digits(n)); vd == vector(#vd, k, vb[k]); \\ Michel Marcus, Apr 08 2019

A038104 Numbers n with property that n is a substring of its base 4 representation.

Original entry on oeis.org

0, 1, 2, 3, 3320, 3321, 3322, 3323, 112000, 112001, 112002, 112003, 121322, 121330, 211320, 211321, 211322, 211323, 320232, 320302, 321230, 321301, 322222, 323221, 323320, 323321, 323322, 323323, 10030333, 20332101, 30132120, 30132121

Offset: 1

Patrick De Geest, Feb 15 1999

			323323 = base 10 -> 1032{323323} = base 4.

Giovanni Resta, Table of n, a(n) for n = 1..107 (terms < 10^16)

Cf. A038102-A038106, A228050-A228052, A227549.

cp's OEIS Frontend

A038102 Numbers k such that k is a substring of its base-2 representation.

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A228050 The decimal representation of n is a substring of its base 7 representation.

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A228052 The decimal representation of n is a substring of its base 9 representation.

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A038103 Numbers k such that k is a substring of its base-3 representation.

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Python

A228051 The decimal representation of n is a substring of its base 8 representation.

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A038105 Numbers n with property that n is a substring of its base 5 representation.

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A307257 Numbers k such that k is the substring identical to the most significant digits of its base-6 representation.

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PARI

A038104 Numbers n with property that n is a substring of its base 4 representation.

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