A063565 -id:A063565 - cp's OEIS frontend

A030000 a(n) is the smallest nonnegative number k such that the decimal expansion of 2^k contains the string n.

10, 0, 1, 5, 2, 8, 4, 15, 3, 12, 10, 40, 7, 17, 18, 21, 4, 27, 30, 13, 11, 18, 43, 41, 10, 8, 18, 15, 7, 32, 22, 17, 5, 25, 27, 25, 16, 30, 14, 42, 12, 22, 19, 22, 18, 28, 42, 31, 11, 32, 52, 9, 19, 16, 25, 16, 8, 20, 33, 33, 23, 58, 18, 14, 6, 16, 46, 24, 15, 34, 29, 21, 17, 30

Offset: 0

N. J. A. Sloane

a(n) is well-defined for all n, because 2^k can actually start with (not just contain) any finite sequence of digits without leading zeros. This follows from the facts that log_10(2) is irrational and that the set of fractional parts of n*x is dense in [0,1] if x is irrational. - Pontus von Brömssen, Jul 21 2021

			2^12 = 4096 is first power of 2 containing a 9, so a(9) = 12.

Giovanni Resta, Table of n, a(n) for n = 0..10000 (terms up to a(1000) from T. D. Noe, a(9634) corrected by Piotr Idzik)

Cf. A030001 (the actual powers of 2), A063565, A018856, A000079, A372044, A372045.

A006889 Exponent of least power of 2 having n consecutive 0's in its decimal representation.

Original entry on oeis.org

0, 10, 53, 242, 377, 1491, 1492, 6801, 14007, 100823, 559940, 1148303, 4036338, 4036339, 53619497, 119476156, 146226201, 918583174, 4627233991, 11089076233

Offset: 0

P. D. Mitchelmore (dh115(AT)city.ac.uk)

A224782(a(n)) = n and A224782(m) <> n for m < a(n). - Reinhard Zumkeller, Apr 30 2013
The name of this sequence previously began "Least power of 2 having exactly n consecutive 0's...". The word "exactly" was unnecessary because the least power of 2 having at least n consecutive 0's in its decimal representation will always have exactly n consecutive 0's. The previous power of two will have had n-1 consecutive 0's with a "5" immediately to the left. - Clive Tooth, Jan 22 2016
a(20) is greater than 12*10^9. - Benjamin Chaffin, Jan 18 2017

			2^53619497 is the smallest power of 2 to contain a run of 14 consecutive zeros in its decimal form.
2^119476156 (a 35965907-digit number) contains the sequence ...40030000000000000008341... about one third of the way through.
2^4627233991 (a 1392936229-digit number) contains the sequence "813000000000000000000538" about 99.5% of the way through. The computation took about six months.

Julian Havil, Impossible?: Surprising Solutions to Counterintuitive Conundrums, Princeton University Press 2008, chapter 15, p. 176ff
Popular Computing (Calabasas, CA), Zeros in Powers of 2, Vol. 3 (No. 25, Apr 1975), page PC25-16 [Gives a(1)-a(8)]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

E. Karst and U. Karst, The first power of 2 with 8 consecutive zeros, Math. Comp. 18 (1964), 508-508.
Popular Computing (Calabasas, CA), Two Tables, Vol. 1, (No. 9, Dec 1973), page PC9-16.

Cf. A031146.

Cf. also A131535, A131536, A259089, A063565, A259091, A259092.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a006889 = fromJust . (`elemIndex` a224782_list)
-- Reinhard Zumkeller, Apr 30 2013

A[0]:= 0:
m:= 1:
for n from 1 while m <= 9 do
  S:= convert(2^n,string);
  if StringTools:-Search(StringTools:-Fill("0",m),S) <> 0 then
    A[m]:= n;
    m:= m+1;
  fi
od:
seq(A[i],i=0..9); # Robert Israel, Jan 22 2016

a = ""; Do[ a = StringJoin[a, "0"]; k = 1; While[ StringPosition[ ToString[2^k], a] == {}, k++ ]; Print[k], {n, 1, 10} ] (* Robert G. Wilson v, edited by Clive Tooth, Jan 25 2016 *)

conseczerorec(n) = my(d=digits(n), i=0, r=0, x=#Str(n)); while(x > 0, while(d[x]==0, i++; x--); if(i > r, r=i); i=0; x--); r
a(n) = my(k=0); while(conseczerorec(2^k) < n, k++); k \\ Felix Fröhlich, Sep 27 2016

3 more terms from Clive Tooth, Jan 24 2001
One more term from Clive Tooth, Nov 28 2001
One more term from Sacha Roscoe (scarletmanuka(AT)iprimus.com.au), Dec 16 2002
a(17) from Sacha Roscoe (scarletmanuka(AT)iprimus.com.au), Feb 06 2007
a(18) from Clive Tooth, Sep 30 2012
Name clarified by Clive Tooth, Jan 22 2016
Definition clarified by Felix Fröhlich, Sep 27 2016
a(19) from Benjamin Chaffin, Jan 18 2017

A176763 Smallest power of 3 whose decimal expansion contains n.

Original entry on oeis.org

59049, 1, 27, 3, 243, 6561, 6561, 27, 81, 9, 10460353203, 1162261467, 129140163, 31381059609, 177147, 1594323, 129140163, 177147, 2187, 19683, 387420489, 2187, 1162261467, 1594323, 243, 2541865828329, 1162261467, 27, 282429536481, 729, 43046721, 531441, 1594323

Offset: 0

Jonathan Vos Post, Apr 25 2010

This is to 3 as A030001 is to 2.

			a(1) = 1 because 3^0 = 1 has "1" as a substring (not a proper substring, though).
a(2) = 27 because 3^3 = 27 has "2" as a substring.
a(10) = 10460353203 because 3^21 = 10460353203 is the smallest power of 3 whose decimal expansion contains "10" (in this case, "10" happens to be the left-hand or initial digits, but that is not generally true).

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A000244, A030000, A030001, A063565, A018856.

A176763[n_] := Block[{k = -1}, While[StringFreeQ[IntegerString[3^++k], IntegerString[n]]]; 3^k]; Array[A176763, 50, 0] (* Paolo Xausa, Apr 03 2024 *)

def a(n):
    k, strn = 0, str(n)
    while strn not in str(3**k): k += 1
    return 3**k
print([a(n) for n in range(33)]) # Michael S. Branicky, Apr 03 2024

a(n) = MIN{A000244(i) such that n in decimal representation is a substring of A000244(i)}.

More terms from Sean A. Irvine and Jon E. Schoenfield, May 05 2010

a(0) prepended by Paolo Xausa, Apr 03 2024

A131535 Exponent of least power of 2 having exactly n consecutive 1's in its decimal representation.

Original entry on oeis.org

1, 0, 40, 42, 313, 485, 1841, 8923, 8554, 81783, 165742, 1371683, 1727601, 9386566, 28190643, 63416789

Offset: 0

Shyam Sunder Gupta, Aug 26 2007

			a(3)=42 because 2^42(i.e. 4398046511104) is the smallest power of 2 to contain a run of 3 consecutive ones in its decimal form.

Popular Computing (Calabasas, CA), Two Tables, Vol. 1, (No. 9, Dec 1973), page PC9-16.

Cf. A006889, A131535, A131536, A259089, A063565, A259091, A259092.

a = ""; Do[ a = StringJoin[a, "1"]; b = StringJoin[a, "1"]; k = 1; While[ StringPosition[ ToString[2^k], a] == {} || StringPosition[ ToString[2^k], b] != {}, k++ ]; Print[k], {n, 1, 10} ]

def A131535(n):
    s, t, m, k, u = '1'*n, '1'*(n+1), 0, 1, '1'
    while s not in u or t in u:
        m += 1
        k *= 2
        u = str(k)
    return m # Chai Wah Wu, Jan 28 2020

2 more terms from Sean A. Irvine, Jul 19 2010
a(13)-a(14) from Lars Blomberg, Jan 24 2013
a(15) from Bert Dobbelaere, Feb 25 2019
a(0) added and a(1) corrected by Chai Wah Wu, Jan 28 2020

A259089 Least k such that 2^k has at least n consecutive 2's in its decimal representation.

Original entry on oeis.org

0, 1, 43, 43, 314, 314, 2354, 8555, 13326, 81784, 279272, 865356, 1727602, 1727602

Offset: 0

N. J. A. Sloane, Jun 18 2015

			a(3)=43 because 2^43 (i.e. 8796093022208) is the smallest power of 2 to contain a run of 3 consecutive twos in its decimal form.

Popular Computing (Calabasas, CA), Two Tables, Vol. 1, (No. 9, Dec 1973), page PC9-16.

Cf. A006889, A131535, A131536, A063565, A259091, A259092.

Table[k = 0; While[! SequenceCount[IntegerDigits[2^k], ConstantArray[2, n]] > 0, k++]; k, {n, 10}] (* Robert Price, May 17 2019 *)

def A259089(n):
    s, k, k2 = '2'*n, 0, 1
    while True:
        if s in str(k2):
            return k
        k += 1
        k2 *= 2 # Chai Wah Wu, Jun 19 2015

a(7)-a(13) from Chai Wah Wu, Jun 20 2015
Definition corrected by Manfred Scheucher, Jun 23 2015
a(0) prepended by Chai Wah Wu, Jan 28 2020

A259091 Smallest k such that 2^k contains two adjacent copies of n in its decimal expansion.

Original entry on oeis.org

53, 40, 43, 25, 18, 16, 46, 24, 19, 33, 378, 313, 170, 374, 361, 359, 64, 34, 507, 151, 348, 246, 314, 284, 349, 314, 261, 151, 385, 166, 156, 364, 65, 219, 371, 359, 503, 148, 155, 352, 349, 308, 247, 255, 192, 387, 165, 149, 171, 150, 210, 155, 209, 101, 505

Offset: 0

N. J. A. Sloane, Jun 18 2015

The multi-digit generalization of A171132. - R. J. Mathar, Jul 06 2015

			2^53 = 9007199254740992 contains two adjacent 0's.

Chai Wah Wu, Table of n, a(n) for n = 0..1000
Popular Computing (Calabasas, CA), Two Tables, Vol. 1, (No. 9, Dec 1973), page PC9-16.

Cf. A006889, A131535, A131536, A259089, A063565, A259092.

Table[k = 0; While[! SequenceCount[IntegerDigits[2^k], Flatten[ConstantArray[IntegerDigits[n], 2]]] > 0, k++]; k, {n, 0, 100}] (* Robert Price, May 17 2019 *)

def A259091(n):
    s, k, k2 = str(n)*2, 0, 1
    while True:
        if s in str(k2):
            return k
        k += 1
        k2 *= 2 # Chai Wah Wu, Jun 18 2015

More terms from Chai Wah Wu, Jun 18 2015

A259092 Smallest k such that 2^k contains three adjacent copies of n in its decimal expansion.

Original entry on oeis.org

242, 42, 43, 83, 44, 41, 157, 24, 39, 50, 949, 1841, 3661, 1798, 1701, 1161, 1806, 391, 1890, 2053, 950, 1164, 2354, 1807, 3816, 1800, 1799, 818, 1702, 2115, 904, 1798, 1807, 2270, 392, 1699, 3022, 394, 2054, 1758, 1804, 2300, 2720, 2403, 3396, 1133, 1808, 3820

Offset: 0

N. J. A. Sloane, Jun 18 2015

The multi-digit generalization of A171242. - R. J. Mathar, Jul 06 2015

			2^242 = 7067388259113537318333190002971674063309935587502475832486424805170479104 contains three adjacent 0's.

Chai Wah Wu, Table of n, a(n) for n = 0..1000
Popular Computing (Calabasas, CA), Two Tables, Vol. 1, (No. 9, Dec 1973), page PC9-16.

Cf. A006889, A131535, A131536, A259089, A063565, A259091.

Table[k = 0; While[! SequenceCount[IntegerDigits[2^k], Flatten[ConstantArray[IntegerDigits[n], 3]]] > 0, k++]; k, {n, 0, 50}] (* Robert Price, May 17 2019 *)

def A259092(n):
    s, k, k2 = str(n)*3, 0, 1
    while True:
        if s in str(k2):
            return k
        k += 1
        k2 *= 2 # Chai Wah Wu, Jun 18 2015

More terms from Chai Wah Wu, Jun 18 2015

A186774 Smallest power of n whose decimal expansion contains n+1, or 0 if no such number exists.

Original entry on oeis.org

32, 243, 256, 625, 7776, 16807, 4096, 31381059609, 0, 121, 79496847203390844133441536, 51185893014090757, 155568095557812224, 22168378200531005859375, 17592186044416, 118587876497, 11019960576, 42052983462257059

Offset: 2

Jonathan Vos Post, Feb 26 2011

More precisely: smallest power of n (with positive integer exponent) whose decimal expansion contains n+1 as a substring of consecutive decimal digits. This is A[n,n+1], the diagonal above the trivial main diagonal of the array A[k,n] = Smallest power of k whose decimal expansion contains n.
The k=2 row A[2,n] = A030001.
The k=3 row A[3,n] = A176763.
The k=4 row A[4,n] = A176764.
The k=5 row A[5,n] = A176765...
a(10^k+1) = (10^k+1)^2 for k > 0. - Chai Wah Wu, Feb 13 2017

			a(2) = 32 = A030001(3) = smallest power of 2 whose decimal expansion contains 3.
a(3) = 243 = A176763(4) = smallest power of 3 whose decimal expansion contains 4.

Chai Wah Wu, Table of n, a(n) for n = 2..1999

Cf. A018856, A063565, A030001, A176763-A176773.

a:= proc(n) local t, k;
      if type(simplify(log[10](n)), integer) then 0
    else t:= cat(n+1);
         for k from 2 while searchtext(t, cat(n^k))=0
         do od; n^k
      fi
    end:
seq(a(n), n=2..40);  # Alois P. Heinz, Feb 26 2011

def A186774(n):
    if sum(int(d) for d in str(n)) == 1:
        return 0
    sn, k = str(n+1), 1
    while sn not in str(k):
        k *= n
    return k # Chai Wah Wu, Feb 13 2017

A371904 a(1) = 1; for n > 1, a(n) is the smallest unused positive number such that 2^a(n) contains a(n-1) as a substring.

Original entry on oeis.org

1, 4, 2, 5, 8, 3, 14, 18, 30, 22, 43, 25, 41, 44, 38, 23, 58, 33, 63, 55, 16, 24, 10, 17, 27, 15, 21, 31, 49, 32, 45, 28, 7, 20, 11, 40, 12, 9, 13, 37, 47, 36, 29, 60, 59, 35, 52, 19, 53, 79, 34, 65, 42, 50, 54, 39, 61, 62, 66, 46, 64, 6, 26, 71, 48, 51, 57, 56, 68, 75, 67, 78, 74, 70, 69, 72

Offset: 1

Scott R. Shannon, Apr 11 2024

The sequence is conjectured to be a permutation of the positive integers.

			a(2) = 4 as 2^4 = 16 which contains a(n-1) = a(1) = '1' as a substring.
a(7) = 14 as 2^14 = 16384 which contains a(n-1) = a(6) = '3' as a substring. Note that 2^5 = 32 also contains '3' as a substring but 5 has already been used.

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Cf. A000079, A063565, A371918, A368866.

A371918 The smallest unused positive number such that 2^a(n) contains n as a substring.

Original entry on oeis.org

10, 4, 1, 5, 2, 8, 6, 15, 3, 12, 17, 40, 7, 27, 18, 21, 14, 34, 30, 13, 11, 24, 43, 41, 19, 50, 28, 38, 47, 32, 22, 49, 25, 63, 33, 35, 16, 37, 23, 42, 53, 44, 59, 45, 46, 52, 60, 31, 20, 39, 54, 9, 51, 29, 74, 57, 48, 56, 61, 69, 55, 58, 66, 76, 26, 73, 72, 36, 62, 82, 64, 70, 77, 67, 65, 91, 78

Offset: 0

Scott R. Shannon, Apr 12 2024

The sequence is conjectured to be a permutation of the positive integers.

			a(0) = 10 as 2^10 = 1024 which contains '0' as a substring.
a(6) = 6 as 2^6 = 64 which contains '6' as a substring. Note that 2^4 also contains '6' but 4 has already been used. This is the first term to differ from A063565.

Scott R. Shannon, Table of n, a(n) for n = 0..10000

Cf. A000079, A371904, A063565, A368866.

cp's OEIS Frontend

A030000 a(n) is the smallest nonnegative number k such that the decimal expansion of 2^k contains the string n.

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A006889 Exponent of least power of 2 having n consecutive 0's in its decimal representation.

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A176763 Smallest power of 3 whose decimal expansion contains n.

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A131535 Exponent of least power of 2 having exactly n consecutive 1's in its decimal representation.

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A259089 Least k such that 2^k has at least n consecutive 2's in its decimal representation.

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A259091 Smallest k such that 2^k contains two adjacent copies of n in its decimal expansion.

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A259092 Smallest k such that 2^k contains three adjacent copies of n in its decimal expansion.