A098174 -id:A098174 - cp's OEIS frontend

A000030 Initial digit of n.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane, Simon Plouffe

When n - a(n)*10^[log_10 n] >= 10^[(log_10 n) - 1], where [] denotes floor, or when n < 100 and 10|n, n is the concatenation of a(n) and A217657(n). - Reinhard Zumkeller, Oct 10 2012, improved by M. F. Hasler, Nov 17 2018, and corrected by Glen Whitney, Jul 01 2022

Equivalent definition: The initial a(0) = 0 is followed by each digit in S = {1,...,9} once. Thereafter, repeat 10 times each digit in S. Then, repeat 100 times each digit in S, etc.

			23 begins with a 2, so a(23) = 2.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David W. Wilson, Table of n, a(n) for n = 0..10000
A. Cobham, Uniform Tag Sequences, Mathematical Systems Theory, 6 (1972), 164-192.

Cf. A010879 (final digit of n).
Cf. A061681, A130571, A109453, A134010, A052038.
Cf. A002993, A089951, A002994, A143464, A098174, A098175, A072543, A072544, A073600, A073601, A037904. - Reinhard Zumkeller, Aug 17 2008

a000030 = until (< 10) (`div` 10) -- Reinhard Zumkeller, Feb 20 2012, Feb 11 2011

[Intseq(n)[#Intseq(n)]: n in [1..100]]; // Vincenzo Librandi, Nov 17 2018

A000030 := proc(n)
    if n = 0 then
        0;
    else
        convert(n,base,10) ;
        %[-1] ;
    end if;
end proc:
seq(A000030(n),n=0..200) ;# N. J. A. Sloane, Feb 10 2017

Join[{0},First[IntegerDigits[#]]&/@Range[90]] (* Harvey P. Dale, Mar 01 2011 *)
Table[Floor[n/10^(Floor[Log10[n]])], {n, 1, 50}] (* G. C. Greubel, May 16 2017 *)
Table[NumberDigit[n,IntegerLength[n]-1],{n,0,100}] (* Harvey P. Dale, Aug 29 2021 *)

a(n)=if(n<10,n,a(n\10)) \\ Mainly for illustration.

A000030(n)=n\10^logint(n+!n,10) \\ Twice as fast as a(n)=digits(n)[1]. Before digits() was added in PARI v.2.6.0 (2013), one could use, e.g., Vecsmall(Str(n))[1]-48. - M. F. Hasler, Nov 17 2018

def a(n): return int(str(n)[0])
print([a(n) for n in range(85)]) # Michael S. Branicky, Jul 01 2022

a(n) = [n / 10^([log_10(n)])] where [] denotes floor and log_10(n) is the logarithm is base 10. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001

a(n) = k for k*10^j <= n < (k+1)*10^j for some j. - M. F. Hasler, Mar 23 2015

A371706 a(n) is the least k > 0 such that n^k contains the digit 1.

Original entry on oeis.org

1, 4, 4, 2, 3, 3, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 3, 3, 3, 3, 3, 3, 3, 2, 4, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 1, 3, 3, 2, 3, 2, 3, 3, 2, 3, 1, 4, 4, 3, 4, 4, 4, 3, 2, 4, 1, 2, 3, 5, 3, 4, 4, 4, 2, 3, 1, 3, 3, 4, 3, 4, 4, 3, 2, 2, 1, 4, 4, 6, 4, 2, 3, 3, 2

Offset: 1

Robert Israel, Apr 03 2024

First n such that a(n) = k: 1, 4, 5, 2, 74, 94, 305, 2975 for k = 1 to 8.
a(n) <= 8 for n <= 240000000.  Is a(n) ever greater than 8?
a(n) <= 8 for n <= 10^11. Moreover the only n <= 10^11 with a(n) = 8 are 2975 * 10^j. - Robert Israel, Apr 05 2024
a(n) <= 8 for n <= 10^13, and a(n) <= 17 for all n; see A275533. - Michael S. Branicky, Apr 08 2024

			a(3) = 4 because 3^1, 3^2 = 9 and 3^3 = 27 have no 1's, but 3^4 = 81 does have a 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A098174, A255430, A255357, A275533.

g:= proc(n) local k;
   for k from 1 do if member(1,convert(n^k,base,10)) then return k fi od;
end proc:
map(g, [$1..100]);

seq={};Do[k=1;While[!ContainsAny[IntegerDigits[n^k],{1}],k++];AppendTo[seq,k],{n,99}];seq (* James C. McMahon, Apr 05 2024 *)

from itertools import count
def A371706(n):
    m = n
    for k in count(1):
        if '1' in str(m):
            return k
        m *= n # Chai Wah Wu, Apr 04 2024

a(n) <= A098174(n).

A098175 Smallest power of n with initial digit = 1 in decimal representation.

Original entry on oeis.org

1, 16, 19683, 16, 125, 1296, 16807, 16777216, 1853020188851841, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 160000, 194481, 10648, 12167, 13824, 15625, 17576, 19683, 17210368, 17249876309, 19683000000000

Offset: 1

Reinhard Zumkeller, Aug 30 2004

A000030(a(n)) = 1; a(n) = n^A098174(n).

f[n_] := Block[{k = 1}, While[IntegerDigits[n^k][[1]] != 1, k++ ]; n^k]; Table[ f[n], {n, 30}] (* Robert G. Wilson v, Sep 01 2004 *)

A309736 a(1) = 1, and for any n > 1, a(n) is the least k > 0 such that the binary representation of n^k starts with "10".

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 2, 3, 5, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 5, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 6, 10, 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Rémy Sigrist, Aug 14 2019

The sequence is well defined; for any n > 0:
- if n is a power of 2, then a(n) = 1,
- if n is not a power of 2, then log_2(n) is irrational,
  hence the function k -> frac(k * log_2(n)) is dense in the interval [0, 1]
  according to Weyl's criterion,
  so for some k > 0, k*log_2(n) = m + 1 + e where m is a positive integer
                                              and 0 <= e < log_2(3) - 1 < 1,
- hence 2 * 2^m <= n^k < 3 * 2^m and a(n) <= k, QED.

			For n = 7:
- the first powers of 7, in decimal as well as in binary, are:
    k  7^k  bin(7^k)
    -  ---  ---------
    1    7        111
    2   49     110001
    3  343  101010111
- hence a(7) = 3.

Rémy Sigrist, Scatterplot of (x, y) such that the binary representation of x^y starts with "10" and x = 2..1024 and y = 1..1024
Eric Weisstein's World of Mathematics, Weyl's Criterion

Cf. A004754, A090996, A098174.

a(n) = { my (nk=n); for (k=1, oo, if (binary(2*nk)[2]==0, return (k), nk *= n)) }

a(n) = 1 iff n belongs to A004754.
a(2*n) = a(n).
A090996(n^a(n)) = 1.

A363683 Square array A(n, k), n, k > 0, read by antidiagonals; A(n, k) is the least e > 0 such that n^e and k^e have the same initial digit, or -1 if no such e exists.

Original entry on oeis.org

1, 4, 4, 9, 1, 9, 2, 17, 17, 2, 3, 7, 1, 7, 3, 4, 5, 8, 8, 5, 4, 5, 4, 9, 1, 9, 4, 5, 8, 2, 3, 11, 11, 3, 2, 8, 16, 5, 11, 6, 1, 6, 11, 5, 16, 1, 17, 7, 4, 9, 9, 4, 7, 17, 1, 1, 4, 17, 10, 6, 1, 6, 10, 17, 4, 1, 1, 4, 9, 14, 5, 15, 15, 5, 14, 9, 4, 1

Offset: 1

Rémy Sigrist, Jun 15 2023

Conjecture: all terms are positive.

			Array A(n, k) begins:
  n\k |  1   2   3   4   5   6   7   8   9  10  11  12
  ----+-----------------------------------------------
    1 |  1   4   9   2   3   4   5   8  16   1   1   1
    2 |  4   1  17   7   5   4   2   5  17   4   4   9
    3 |  9  17   1   8   9   3  11   7  17   9  16   5
    4 |  2   7   8   1  11   6   4  10  14   2   2   2
    5 |  3   5   9  11   1   9   6   5   4   3   3   3
    6 |  4   4   3   6   9   1  15   7  11   4   4  10
    7 |  5   2  11   4   6  15   1  17  18   5   5   4
    8 |  8   5   7  10   5   7  17   1  18   8  28   6
    9 | 16  17  17  14   4  11  18  18   1  16  23   8
   10 |  1   4   9   2   3   4   5   8  16   1   1   1
   11 |  1   4  16   2   3   4   5  28  23   1   1   1
   12 |  1   9   5   2   3  10   4   6   8   1   1   1

Rémy Sigrist, Colored representation of the table for n, k <= 1200

Cf. A000030, A073601, A088995, A098174, A359698.

A(n,k) = { for (e = 1, oo, if (digits(n^e)[1]==digits(k^e)[1], return (e));); }

A(n, k) = A(k, n).
A(10*n, k) = A(n, k).
A(n, n) = 1.
A(n, 1) = A098174(n).

A367821 a(n) = first exponent k such that n^k starts with 9 or -1 if n is a power of 10.

Original entry on oeis.org

-1, 53, 2, 78, 10, 176, 13, 21, 1, -1, 24, 25, 26, 34, 17, 39, 13, 39, 25, 53, 3, 32, 11, 21, 5, 12, 37, 29, 41, 2, 2, 89, 25, 15, 11, 88, 7, 12, 5, 78, 13, 8, 11, 45, 3, 3, 55, 22, 13, 10, 24, 60, 11, 15, 4, 4, 37, 17, 22, 176, 14, 5, 5, 26, 43, 39, 6, 6, 25, 13, 7

Offset: 1

William Hu, Dec 01 2023

First k such that log_10(9) <= fractional part of k*log_10(n) < 1.

For a non-power of 10, the leading digits of the powers of n follow Benford's law, therefore making 9 the least common leading digit of powers of n.

			a(2) = 53 because smallest power of 2 that starts with 9 is 2^53 = 9007199254740992. See A018856.
a(3) = 2 because 3^2 = 9.

Cf. A018856, A098174, A367854 (record high indices).

a(n) = my(t); if ((n==1) || (n==10) || (ispower(n,,&t) && (t==10)), -1, my(k=1); while (digits(n^k)[1] != 9, k++); k); \\ Michel Marcus, Dec 03 2023

def a(n: int) -> int:
    s = str(n)
    if n <= 1 or (s[0] == '1' and set(s[1:]) == {'0'}):
        return -1
    pwr, e = 1, 0
    while str(pwr)[0] != '9':
        pwr *= n
        e += 1
    return e

cp's OEIS Frontend

A000030 Initial digit of n.

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A371706 a(n) is the least k > 0 such that n^k contains the digit 1.

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A098175 Smallest power of n with initial digit = 1 in decimal representation.

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A309736 a(1) = 1, and for any n > 1, a(n) is the least k > 0 such that the binary representation of n^k starts with "10".

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A363683 Square array A(n, k), n, k > 0, read by antidiagonals; A(n, k) is the least e > 0 such that n^e and k^e have the same initial digit, or -1 if no such e exists.

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A367821 a(n) = first exponent k such that n^k starts with 9 or -1 if n is a power of 10.

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