A038102 -id:A038102 - cp's OEIS frontend

A350572 Base-10 version of A038102 interpreted as base 2.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 16, 17, 19, 20, 21, 23, 24, 25, 28, 29, 32, 33, 40, 42, 44, 45, 47, 48, 49, 53, 60, 61, 64, 65, 66, 80, 81, 82, 90, 91, 96, 97, 100, 112, 113, 121, 128, 129, 137, 144, 145, 152, 161, 192, 193, 203

Offset: 1

Robert G. Wilson v, Jun 30 2014; entered by Hans Havermann, Jan 06 2022

Hans Havermann, Table of n, a(n) for n = 1..1512 (terms 1..1167 from Robert G. Wilson v).

Cf. A038102, A350573.

B2[n_] := FromDigits[IntegerDigits[n, 2]]
SP[n_] := (b = B2[n]; StringPosition[ToString[B2[b]], ToString[b]])
n=0; t={}; While[n<10^6, If[SP[n]!={}, AppendTo[t,n]]; n++]; t
(* Hans Havermann, Jan 06 2022 *)

def ok(n): b = bin(n)[2:]; return b in bin(int(b))
print([k for k in range(204) if ok(k)]) # Michael S. Branicky, Jan 06 2022

A181929 Numbers n such that n is the substring identical to the most significant bits of its base 2 representation.

Original entry on oeis.org

0, 1, 10, 110, 10011, 110101, 10011000, 110100011, 10010101001, 101111000101, 110011001110, 10010001101010, 101101111110011, 110010000001101, 1111110010100011, 10001110000111111, 101100111001011100, 110000110110011001, 1111011010110001101

Offset: 1

Douglas Latimer, Apr 02 2012

The main idea behind my program is that if we say start searching from 10000, which is 10011... binary, then as the binary string for the first 5 places is larger than our decimal value, then the decimal value can be immediately jumped to 10011 for the next search number. Repeating this process (while also doing slight jumps if the decimal value is larger than the binary), allows ones to do very large jumps in the checked decimal values, sometimes eliminating an entire string of length n in just a few checks. I got the idea from similar people were using when searching for the next term in A258107. - Scott R. Shannon, Feb 25 2021

			The number 110 is represented in the binary system by the string "1101110". 110 is a three-digit number, so we consider the 3 most significant bits, which are "110", identical to the string of digits used to represent the number 110. Thus 110 is in the sequence.

Scott R. Shannon, Table of n, a(n) for n = 1..1000

This is a subsequence of A038102. Sequence A181891 has a similar definition.

Subsequence of A007088.

fQ[n_] := Module[{d = IntegerDigits[n], len}, len = Length[d]; d == Take[IntegerDigits[n, 2], len]]; Select[Range[0, 1000000], fQ] (* T. D. Noe, Apr 03 2012 *)

{for(vv=0,2000000,bvv=binary(vv);
ll=length(bvv);texp=0;btod=0;
forstep(i=ll,1,-1,btod=btod+bvv[i]*10^texp;texp++);
bigb=binary(btod);swsq=1;
for(j=1,ll,if(bvv[j]!=bigb[j],swsq=0));
if(swsq==1,print(btod)))}

lista(nn) = {for (n=0, nn, if (n==0, print1(n, ", "), my(b = binary(n), db = fromdigits(b), bb = binary(db)); if (vector(#b, k, bb[k]) == b, print1(db, ", "));););} \\ Michel Marcus, Feb 10 2021

A038106 Numbers k with the property that k is a substring of its base-6 representation.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 45240, 45241, 45242, 45243, 45244, 45245, 54000, 54001, 54002, 54003, 54004, 54005, 304200, 304201, 304202, 304203, 304204, 304205, 3240000, 3240001, 3240002, 3240003, 3240004, 3240005, 3544200, 3544201, 3544202

Offset: 1

Patrick De Geest, Feb 15 1999

			304201_10 = 10{304201}_6.

Giovanni Resta, Table of n, a(n) for n = 1..125 (terms < 10^18)

Cf. A038102-A038105, A228050-A228052, A227549.

Select[Range[0,355*10^4],SequenceCount[IntegerDigits[#,6],IntegerDigits[#]]>0&] (* Harvey P. Dale, Mar 11 2023 *)

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.

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.

A181891 Numbers n such that n is the substring identical to the least significant bits of its base 2 representation.

Original entry on oeis.org

0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1100, 1101, 10000, 10001, 10100, 10101, 11000, 11001, 11100, 11101, 100000, 100001, 101100, 101101, 110000, 110001, 111100, 111101, 1000000, 1000001, 1010000, 1010001, 1100000, 1100001, 1110000, 1110001

Offset: 1

Douglas Latimer, Mar 30 2012

Terms with odd index, that is, a(1), a(3), a(5), ... are all multiples of 10. Each even-index term is one more than its predecessor, so that a(2n) = a(2n-1) + 1. [Douglas Latimer, Apr 26 2013]

			The number 11 is represented in the binary system by the string "1011". 11 is a two-digit number, so we consider the 2 least significant bits, which are "11", identical to the string of digits used to represent the number 11. Thus 11 is in the sequence.

Douglas Latimer, Table of n, a(n) for n = 1..850

This is a subsequence of A038102. Sequence A181929 has similar definition.

fQ[n_] := Module[{d = IntegerDigits[n], len}, len = Length[d]; d == Take[IntegerDigits[n, 2], -len]]; Select[Range[0, 1000000], fQ] (* T. D. Noe, Apr 03 2012 *)

{for(vv=0,200,bvv=binary(vv);
ll=length(bvv);texp=0;btod=0;
forstep(i=ll,1,-1,btod=btod+bvv[i]*10^texp;texp++);
bigb=binary(btod);lbb=length(bigb);swsq=1;
forstep(j=ll,1,-1,if(bvv[j]!=bigb[lbb],swsq=0);lbb--);
if(swsq==1,print(btod)))}

cp's OEIS Frontend

A350572 Base-10 version of A038102 interpreted as base 2.

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A181929 Numbers n such that n is the substring identical to the most significant bits of its base 2 representation.

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A038106 Numbers k with the property that k is a substring of its base-6 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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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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A038104 Numbers n with property that n is a substring of its base 4 representation.

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A181891 Numbers n such that n is the substring identical to the least significant bits of its base 2 representation.

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