A004218 -id:A004218 - cp's OEIS frontend

A055642 Number of digits in the decimal expansion of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 0

Henry Bottomley, Jun 06 2000

From Hieronymus Fischer, Jun 08 2012: (Start)
For n > 0 the first differences of A117804.
The total number of digits necessary to write down all the numbers 0, 1, 2, ..., n is A117804(n+1). (End)
Here a(0) = 1, but a different common convention is to consider that the expansion of 0 in any base b > 0 has 0 terms and digits. - M. F. Hasler, Dec 07 2018

			Examples:
999: 1 + floor(log_10(999)) = 1 + floor(2.x) = 1 + 2 = 3 or
      ceiling(log_10(999+1)) = ceiling(log_10(1000)) = ceiling(3) = 3;
1000: 1 + floor(log_10(1000)) = 1 + floor(3) = 1 + 3 = 4 or
      ceiling(log_10(1000+1)) = ceiling(log_10(1001)) = ceiling(3.x) = 4;
1001: 1 + floor(log_10(1001)) = 1 + floor(3.x) = 1 + 3 = 4 or
      ceiling(log_10(1001+1)) = ceiling(log_10(1002)) = ceiling(3.x) = 4;

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A043537, A178788, A046034, A019546, A054899, A122840, A055640, A055641, A102669-A102685, A117804, A160093, A160094, A196563, A196564, A000120, A000788, A023416, A059015 (for base 2).
Cf. A262190, A262198.
Cf. A007953: sum of digits.

a055642 :: Integer -> Int
a055642 = length . show  -- Reinhard Zumkeller, Feb 19 2012, Apr 26 2011

[ #Intseq(n): n in [0..105] ];   //  Bruno Berselli, Jun 30 2011
(Common Lisp) (defun A055642 (n) (if (zerop n) 1 (floor (log n 10)))) ; James Spahlinger, Oct 13 2012

A055642 := proc(n)
        max(1,ilog10(n)+1) ;
end proc:  # R. J. Mathar, Nov 30 2011

Join[{1}, Array[ Floor[ Log[10, 10# ]] &, 104]] (* Robert G. Wilson v, Jan 04 2006 *)
Join[{1},Table[IntegerLength[n],{n,104}]]
IntegerLength[Range[0,120]] (* Harvey P. Dale, Jul 02 2016 *)

a(n)=#Str(n) \\ M. F. Hasler, Nov 17 2008

A055642(n)=logint(n+!n,10)+1 \\ Increasingly faster than the above, for larger n. (About twice as fast for n ~ 10^7.) - M. F. Hasler, Dec 07 2018

def a(n): return len(str(n))
print([a(n) for n in range(121)]) # Michael S. Branicky, May 10 2022

def A055642(n): # Faster than len(str(n)) from ~ 50 digits on
    L = math.log10(n or 1)
    if L.is_integer() and 10**int(L)>n: return int(L or 1)
    return int(L)+1  # M. F. Hasler, Apr 08 2024

a(A046760(n)) < A050252(A046760(n)); a(A046759(n)) > A050252(A046759(n)). - Reinhard Zumkeller, Jun 21 2011
a(n) = A196563(n) + A196564(n).
a(n) = 1 + floor(log_10(n)) = 1 + A004216(n) = ceiling(log_10(n+1)) = A004218(n+1), if n >= 1. - Daniel Forgues, Mar 27 2014
a(A046758(n)) = A050252(A046758(n)). - Reinhard Zumkeller, Jun 21 2011
a(n) = A117804(n+1) - A117804(n), n > 0. - Hieronymus Fischer, Jun 08 2012
G.f.: g(x) = 1 + (1/(1-x))*Sum_{j>=0} x^(10^j). - Hieronymus Fischer, Jun 08 2012
a(n) = A262190(n) for n < 100; a(A262198(n)) != A262190(A262198(n)). - Reinhard Zumkeller, Sep 14 2015

A067175 Number of digits in the n-th primorial (A002110).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 15, 17, 18, 20, 22, 24, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 56, 58, 60, 62, 64, 66, 69, 71, 73, 76, 78, 80, 82, 85, 87, 89, 92, 94, 97, 99, 101, 104, 106, 109, 111, 113, 116, 118, 121, 123, 126, 128, 131, 133

Offset: 0

Lekraj Beedassy, Feb 18 2002

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 5001 terms from Soumyadeep Dhar)

Cf. A002110. Essentially the same as A048856.

Cf. A004216, A004218, A055642, A387173.

with(numtheory): it := 1: for n from 1 to 150 do it := it*ithprime(n): printf(`%d,`,ceil(log[10](it))) od:

Table[Floor[Log[10, Product[Prime[k], {k, n}]] + 1], {n, 0, 67}]
Join[{1},IntegerLength/@FoldList[Times,Prime[Range[70]]]] (* Harvey P. Dale, Jan 03 2024 *)

a(n) = 1 + logint(vecprod(primes(n)), 10) \\ Andrew Howroyd, Apr 24 2021

from sympy import primorial
def a(n): return 1 if n == 0 else len(str(primorial(n)))
print([a(n) for n in range(68)]) # Michael S. Branicky, Apr 24 2021

Edited and extended by Robert G. Wilson v and James Sellers, Feb 19 2002

Offset corrected by Arkadiusz Wesolowski, May 04 2013

A052038 First nonzero digit in expansion of 1/n.

Original entry on oeis.org

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

Offset: 1

Patrick De Geest, Dec 15 1999

The number of times each digit occurs for numbers < 10^k:
...\a(n)==1.........2.......3........4........5........6........7........8........9
10^k\
1.........5.........2........1........0........1........0........0........0........0
2........55........19........9........5........5........2........2........1........1
3.......555.......186.......92.......55.......39.......26.......19.......15.......12
4......5555......1853......925......555......373......264......197......154......123
5.....55555.....18520.....9258.....5555.....3707.....2645.....1982.....1543.....1234
6....555556....185187....92591....55555....37041....26454....19839....15432....12345
7...5555555...1851854...925924...555555...370375...264549...198410...154321...123456
8..55555555..18518521..9259257..5555555..3703709..2645501..1984124..1543210..1234567
9.555555555.185185188.92592590.55555555.37037043.26455025.19841266.15432099.12345678
...
Inf. ...5/9......5/27.....5/54.....5/90.....1/27........?........?........?........?

Cf. A052039, A033330, A033420, A061861.

f[n_] := RealDigits[1/n, 10, 12][[1, 1]]; Array[f, 105]

a(n) = floor(10^floor(1+log_10(n-1))/n). After 10^k terms the number of times m will have appeared will be about 10^(k+1)/(9*m*(m+1)), e.g., 1 will appear just over 55.5% of the time. - Henry Bottomley, May 11 2001

a(n) = A000030(floor(A011557(k)/n)) for k >= A004218(n). - Reinhard Zumkeller, Feb 27 2011

A004230 a(n) = 10000*log_10(n) rounded up.

Original entry on oeis.org

0, 3011, 4772, 6021, 6990, 7782, 8451, 9031, 9543, 10000, 10414, 10792, 11140, 11462, 11761, 12042, 12305, 12553, 12788, 13011, 13223, 13425, 13618, 13803, 13980, 14150, 14314, 14472, 14624

Offset: 1

N. J. A. Sloane

nonn

Cf. A004228, A004239.

Cf. A004218, A004221, A004224, A004227.

A176087 Numbers n such that (10^n-1)/3 * 10^ceiling(log_10(n+1)) + n is prime.

Original entry on oeis.org

1, 253, 35473

Offset: 1

Rick L. Shepherd, Apr 08 2010

No term is a multiple of 2, 3, or 5. The decimal expansion of each corresponding prime is n 3's with n's decimal expansion concatenated. Probable primes found by PrimeForm. Prime for 253 proved by Primo. No more terms up to 50000.

			The first term is 1 because 31 is prime.

Cf. A070746 (n 1's followed by n is prime), A084428 (n 7's followed by n is prime), A174710, A004218.

is(n)=ispseudoprime((10^n-1)/3 * 10^logint(10*n,10) + n) \\ Charles R Greathouse IV, May 22 2017

A004224 a(n) = 100*log_10(n) rounded up.

Original entry on oeis.org

0, 31, 48, 61, 70, 78, 85, 91, 96, 100, 105, 108, 112, 115, 118, 121, 124, 126, 128, 131, 133, 135, 137, 139, 140, 142, 144, 145, 147, 148, 150, 151, 152, 154, 155, 156, 157, 158, 160, 161, 162, 163, 164, 165

Offset: 1

N. J. A. Sloane

nonn

Cf. A004222, A004223.

Cf. A004218, A004221, A004227, A004230.

Ceiling[100 Log10[Range[50]]] (* Harvey P. Dale, Jun 05 2021 *)

A123119 Number of digits in sum of first n primes (A007504).

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Sep 28 2006

Since A007504(n) has the asymptotic expression ~ n^2 * log(n) / 2, a(n) has the asymptotic expression n^2 * log(n) / 2 = floor(log_10(10* n^2 * log(n) / 2)) = floor(log_10(5* n^2 * log(n))) = floor(log_10(5) + log_10(n^2) + log_10(log(n))) = floor(0.698970004 + 2*log_10(n) + log_10(log(n))). What is the smallest n such that a(n) = 5, 6, 7, ...?

			a(3) = 2 because 2 + 3 + 5 = 10 has 2 digits in its decimal expansion.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A000041, A004216, A004218, A034386, A055642, A111287.

f[n_] := Floor[ Log[10, Sum[Prime@i, {i, n}]] + 1]; Array[f, 105] (* Robert G. Wilson v *)
f[n_] := IntegerLength[Total[Prime[Range[n]]]]; Array[f, 105] (* Jan Mangaldan, Jan 04 2017 *)
IntegerLength/@Accumulate[Prime[Range[110]]] (* Harvey P. Dale, Jan 26 2019 *)

a(n) = A055642(A007504(n)) = floor(log_10(10*A007504(n))) = A004216(A007504(n)) + 1 = A004218(A007504(n) + 1).

More terms from Robert G. Wilson v, Oct 05 2006

A298486 Square array T(n, k), n >= 0, k >= 0, read by antidiagonals upwards: T(n, k) = the (k+1)-th nonnegative number m such that n + m can be computed with carry in decimal base.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 0, 1, 20, 11, 11, 11, 11, 11, 11, 11, 11, 11, 0, 1

Offset: 0

Rémy Sigrist, Jan 20 2018

The corresponding sequence for the binary base is A295653.

			Square array begins:
  n\k|  0   1   2   3   4   5   6   7   8   9   10  ...
  ---+-------------------------------------------------
    0|  0   1   2   3   4   5   6   7   8   9   10  ... <-- A001477
    1|  0   1   2   3   4   5   6   7   8  10   11  ...
    2|  0   1   2   3   4   5   6   7  10  11   12  ...
    3|  0   1   2   3   4   5   6  10  11  12   13  ...
    4|  0   1   2   3   4   5  10  11  12  13   14  ...
    5|  0   1   2   3   4  10  11  12  13  14   20  ...
    6|  0   1   2   3  10  11  12  13  20  21   22  ...
    7|  0   1   2  10  11  12  20  21  22  30   31  ...
    8|  0   1  10  11  20  21  30  31  40  41   50  ...
    9|  0  10  20  30  40  50  60  70  80  90  100  ... <-- A008592
   10|  0   1   2   3   4   5   6   7   8   9   10  ...

Rémy Sigrist, Table of n, a(n) for n = 0..5150

Cf. A001477, A004218, A008592, A122840, A295653, A298372.

T(n,k,{base=10}) = my (v=0, p=1); while (k, my (r=base - (n%base)); v += p*(k%r); n \= base; k \= r; p *= base); v

For any n >= 0 and k >= 0:
- T(0, k) = k,
- T(9, k) = 10 * k,
- T(10^n - 1, k) = 10^n * k,
- T(n, 0) = 0,
- T(n, 1) = 10^A122840(n+1),
- T(n, k + A298372(n)) = k + 10^A004218(n+1) (i.e. each row is linear).

A321999 Sum of digits of n minus the number of digits of n.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 5, 6, 7, 8, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 7

Offset: 0

M. F. Hasler, Dec 07 2018

Concerning 0 we use the convention that 0 has 0 digits, so a(0) = 0 - 0 = 0, a(1) = 1 - 1 = 0, and a(10) = 1 - 2 = -1 is the first negative terms of the sequence.

			a(0) = 0 - 0 = 0. (We consider 0 has 0 digits.)
a(1) = 1 - 1 = 0;
a(2) = 2 - 1 = 1, ...,
a(9) = 9 - 1 = 8. (General formula: a(10^k - 1) = 8*k.)
a(10) = 1 - 2 = -1. (General formula: a(10^k) = -k.)
a(11) = 1+1 - 2 = 0, ...,
a(19) = 1+9 - 2 = 8;
a(20) = 2+0 - 2 = 0. (General formula: a(m*10^k) = a(m) - k.)
a(29) = 2+9 - 2 = 9, ...,
a(99) = 9+9 - 2 = 16: cf. a(9);
a(100) = 1+0+0 - 3 = -2;
a(101) = 1+0+1 - 3 = -1;
a(102) = 1+0+2 - 3 = 0, ...,
a(109) = 1+0+9 - 3 = 7;
a(110) = 1+1+0 - 3 = -1, ...,
a(119) = 1+1+9 - 3 = 8, ...,
a(199) = 1+9+9 - 3 = 16,
a(200) = 2+0+0 - 3 = -1: cf. a(20), ...,
a(999) = 9+9+9 - 3 = 24: cf. a(9);
a(1000) = 1+0+0+0 - 4 = -3, ...,
a(1001) = 1+0+0+1 - 4 = -2, ....

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A007953 (digit sum of n), A004218 (ceiling(log_10(n))), A055642 (number of digits of n).

The zeroes of this sequence, except 0 itself, are in A061384.

a:= n-> add(i, i=convert(n, base, 10))-length(n):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 10 2018

Table[(Plus@@IntegerDigits[n]) - Length[IntegerDigits[n]] + KroneckerDelta[n, 0], {n, 0, 99}] (* Alonso del Arte, Dec 07 2018 *)
Table[Total[IntegerDigits[n]]-IntegerLength[n],{n,0,100}] (* Harvey P. Dale, Dec 27 2022 *)

A321999(n)=sumdigits(n)-if(n,logint(n,10)+1)

a(n) = A007953(n) - A004218(n+1) = A007953(n) - A055642(n) for all n > 0. a(m*10^k) = a(m) - k for all m > 0, k >= 0, in particular:
a(10^k) = -k for all k >= 0. a(m) = m - 1 for 0 < m < 10.
a(n+1) = a(n) + 1 unless n = 9 (mod 10), in which case a(n+1) = a((n+1)/10).
a(10^k-1) = 8*k.

cp's OEIS Frontend

A055642 Number of digits in the decimal expansion of n.

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A067175 Number of digits in the n-th primorial (A002110).

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A052038 First nonzero digit in expansion of 1/n.

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A004230 a(n) = 10000*log_10(n) rounded up.

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A176087 Numbers n such that (10^n-1)/3 * 10^ceiling(log_10(n+1)) + n is prime.

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A004224 a(n) = 100*log_10(n) rounded up.

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A123119 Number of digits in sum of first n primes (A007504).

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A298486 Square array T(n, k), n >= 0, k >= 0, read by antidiagonals upwards: T(n, k) = the (k+1)-th nonnegative number m such that n + m can be computed with carry in decimal base.

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