A018856 -id:A018856 - 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.

A018802 Smallest power of 2 that begins with n.

Original entry on oeis.org

1, 2, 32, 4, 512, 64, 70368744177664, 8, 9007199254740992, 1024, 1125899906842624, 128, 131072, 140737488355328, 151115727451828646838272, 16, 17179869184, 18014398509481984, 19342813113834066795298816, 2048, 2147483648

Offset: 1

David W. Wilson

			a(7) = 70368744177664, because 2^46 is the smallest power of 2 that begins with a 7.

A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems With Elementary Solutions, Vol. 1 pp. 29, 199-200; Prob. 91a, Dover NY 1987

Christian N. K. Anderson, Table of n, a(n) for n = 1..981 (a(982) has 1196 digits)

Cf. A018856.

Cf. this sequence (k=2), A018857 (k=3), A018859 (k=4), A018861 (k=5), A018863 (k=6), A018865 (k=7), A018867 (k=8), A018869 (k=9).

library(gmp); ndig<-function(i) nchar(as.character(as.bigz(i))) y=as.bigz(rep(0,100)); for(i in 1:100) {n=as.bigz(2); while(substr(n,1,ndig(i))!=as.character(i)) n=n*2; y[i]=n; } # Christian N. K. Anderson, May 23 2013

a(n) = 2^A018856(n). - Seiichi Manyama, Jan 29 2017

a(2^n) = 2^n for n >= 0. - Seiichi Manyama, Jan 29 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

A363060 Numbers k such that 5 is the first digit of 2^k.

Original entry on oeis.org

9, 19, 29, 39, 49, 59, 69, 102, 112, 122, 132, 142, 152, 162, 172, 195, 205, 215, 225, 235, 245, 255, 265, 298, 308, 318, 328, 338, 348, 358, 391, 401, 411, 421, 431, 441, 451, 461, 494, 504, 514, 524, 534, 544, 554, 587, 597, 607, 617, 627, 637, 647, 657, 680, 690

Offset: 1

Ctibor O. Zizka, May 16 2023

The asymptotic density of this sequence is log_10(6/5) = 0.0791812... . - Amiram Eldar, May 16 2023

In base B we may consider numbers k such that some integer Y >= 1 forms the first digit(s) of X^k. For such numbers k the following inequality holds: log_B(Y) - floor(log_B(Y)) <= k*log_B(X) - floor(k*log_B(X)) < log_B(Y+1) - floor(log_B(Y+1)). The irrationality of log_B(X) is the necessary condition; see the Links section. Examples in the OEIS: B = 10, X = 2; Y = 1 (A067497), Y = 2 (A067469), Y = 3 (A172404).

			k = 9: the first digit of 2^9 = 512 is 5, thus k = 9 is a term.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
P. Bohl, Über ein in der Theorie der säkularen Störungen vorkommendes Problem, J. Reine Angew. Math. 135 (1909), 189-283.
Luboš Pick, On One Application of Irrationality in Search of the Seventh Heaven, Pokroky matematiky, fyziky a astronomie, vol. 67 (2022), 37-44.
Hermann Weyl, Über die gibbs’sche Erscheinung und verwandte Konvergenzphänomene, Rend. Circ. Matem. Palermo 30 (1910), 377-407.
Hermann Weyl, Über die Gleichverteilung von Zahlen mod. Eins., Mathematische Annalen 77 (1916), 313-352.

Cf. A000079, A008952, A018856.

Cf. A067497, A067469, A172404, A367294, A367295, A330243, A367296, A097415.

R:= NULL: count:= 0: t:= 1:
for k from 1 while count < 100 do
  t:= 2*t;
  if floor(t/10^ilog10(t)) = 5 then R:= R,k; count:= count+1 fi
od:
R; # Robert Israel, May 19 2023

Select[Range[700], IntegerDigits[2^#][[1]] == 5 &] (* Amiram Eldar, May 16 2023 *)

isok(k) = digits(2^k)[1] == 5; \\ Michel Marcus, May 16 2023

from itertools import count, islice
def A363060_gen(startvalue=1): # generator of terms >= startvalue
    m = 1<<(k:=max(startvalue,1))
    for i in count(k):
        if str(m)[0]=='5':
            yield i
        m <<= 1
A363060_list = list(islice(A363060_gen(),20)) # Chai Wah Wu, May 21 2023

A152561 Smallest m such that 2^m begins with a 1 and n 0's.

Original entry on oeis.org

0, 10, 196, 2136, 15437, 70777, 325147, 6432163, 198096465, 15042141867, 76590952427, 82361153417, 578451474249, 6198243909905, 410654730169643, 2178891522315645, 11458333085072746, 112404439328411815, 2539468772213180761, 4415969241540963378, 218809557168391974468

Offset: 0

Jon E. Schoenfield, Dec 07 2008

			2^a(0) = 2^0     = 1
2^a(1) = 2^10    = 1024
2^a(2) = 2^196   = 1004336277661868922...
2^a(3) = 2^2136  = 1000162894137615530...
2^a(4) = 2^15437 = 1000099165462017237...
2^a(5) = 2^70777 = 1000007160137454597...

Zhao Hui Du, Table of n, a(n) for n = 0..500

Cf. A000079, A011557, A018856.

a(n) = A018856(10^n). - Andrew Howroyd, Sep 30 2024

a(0)=0 prepended by Pontus von Brömssen, May 11 2021

a(18) and above from Zhao Hui Du, Oct 03 2024

A180690 11^a(n) is smallest power of 11 beginning with n.

Original entry on oeis.org

0, 8, 12, 15, 17, 19, 21, 22, 24, 25, 1, 2, 3, 4, 29, 5, 6, 55, 7, 56, 8, 57, 9, 82, 10, 107, 59, 11, 84, 36, 12, 85, 61, 13, 110, 62, 14, 135, 87, 63, 15, 136, 88, 64, 16, 137, 113, 89, 41, 17, 138, 114, 66, 42, 18, 139, 115, 91, 67, 43, 19, 140, 116, 92, 68, 44, 20, 141, 117, 93

Offset: 1

Daniel Mondot, Sep 17 2010

D. Mondot, Table of n, a(n) for n=1..32699

Cf. A018856, A018870, A180689, A180692.

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

A371887 a(1) = 1; for n > 1, a(n) is the smallest positive integer k such that the digits of 2^k contain 2^a(n-1) as a proper substring.

Original entry on oeis.org

1, 5, 15, 507

Offset: 1

Adam Vulic, Apr 11 2024

From David A. Corneth, Apr 11 2024: (Start)
This sequence is well defined as A030000 is well defined; every finite string of digits is contained in some power of 2.
An upper bound for a(n), n > 1, can be found by solving 2^k == 2^a(n-1) (mod 10^m) where m is the number of digits of 2^a(n-1) (cf. A034887). This gives a(n) <= k = a(n-1) + 4*5^(m-1) (cf. A005054). So a(5) <= 507 + 4*5^152, which is about 7*10^106. (End)

			a(2) is the smallest k > 0 such that the digits of 2^k contain 2^a(1) = 2^1 = 2 as a proper substring, so a(2) = 5. (2^5 = 32.)
a(3) is the smallest k > 0 such that the digits of 2^k contain 2^a(2) = 32 as a proper substring, so a(3) = 15. (2^15 = 32768.)

Brady Haran, Apocalyptic Numbers, Numberphile video, 2024.

Cf. A018856, A005054, A030000, A034887.

k = 0; Rest@ NestList[(While[SequenceCount[IntegerDigits[2^k], IntegerDigits[2^#]] == 0, k++]; k++; k - 1) &, 1, 4] (* Michael De Vlieger, Apr 19 2024 *)

A374923 a(n) is the least k such that 2^k begins with n!.

Original entry on oeis.org

0, 0, 1, 6, 81, 80, 56, 7284, 33889, 2044921, 8151937, 127668791, 258943304, 19207561921, 189815680859, 2687562198191, 75909586168557, 512148453482307, 5376323935222903, 502774568129731130, 1053338431686717460, 122114339415457901831, 2120280158164651048122

Offset: 0

Zhining Yang, Jul 23 2024

			a(4) = 81 because 2^81 = 2417851639229258349412352 is the smallest power of 2 beginning with 4! = 24.

Zhao Hui Du, Table of n, a(n) for n = 0..100

Cf. A018856, A000142, A152561, A374922.

a[n_] := Module[{target = IntegerDigits[n!], k = 0},
   While[UnsameQ[Take[IntegerDigits[2^k], Length@target], target],
    k++]; k];
Table[a[n], {n, 0, 8}]

a(n) = A018856(n!).

a(13) onwards from Zhao Hui Du, Oct 02 2024

A363873 Least k such that 2^k begins with n but is not exactly n.

Original entry on oeis.org

4, 8, 5, 12, 9, 6, 46, 13, 53, 10, 50, 7, 17, 47, 77, 14, 34, 54, 84, 11, 31, 51, 61, 81, 8, 18, 38, 48, 68, 78, 98, 15, 25, 35, 45, 55, 75, 85, 95, 12, 22, 32, 42, 145, 52, 62, 72, 82, 92, 102, 9, 19, 29, 39, 142, 49, 59, 162, 69, 79, 89, 192, 99, 109, 16, 119, 26, 36, 139, 46

Offset: 1

Robert G. Wilson v, Jul 03 2023

This is not an injective function. a(2) = a(25) = 8.

a(n) > 3.

			a(1) = 4 since 2^4 = 16 starts with 1 and is not 1 itself (the way 2^0 = 1 would be);
a(2) = 8 (not 1: 2^1 = 2) since 2^8 = 256;
a(3) = 5 since 2^5 = 32;
a(4) = 12 (not 2: 2^2 = 4) since 2^12 = 4096;
a(5) = 9 since 2^9 = 512; etc.

Cf. A000079, A018856.

a[n_] := Block[{j = IntegerLength@ n, k = 1}, While[ IntegerLength[2^k] < j || Quotient[2^k, 10^(IntegerLength[2^k] - j)] != n || n == 2^k, k++]; k]; Array[ a, 70]

def A363873(n):
    m, s = 1<<(k:=n.bit_length()-1), str(n)
    while m<=n or not str(m).startswith(s):
        k += 1
        m <<= 1
    return k # Chai Wah Wu, Aug 06 2023

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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A018802 Smallest power of 2 that begins with n.

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A176763 Smallest power of 3 whose decimal expansion contains n.

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A363060 Numbers k such that 5 is the first digit of 2^k.

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A152561 Smallest m such that 2^m begins with a 1 and n 0's.

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A180690 11^a(n) is smallest power of 11 beginning with n.

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

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