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

Author

Henry Bottomley, Jun 06 2000

Keywords

• base
• easy
• nonn
• nice

Comments

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

			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;
		

Links

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

Crossrefs

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.

Programs

Haskell

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

Magma

[ #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

Maple

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

Mathematica

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 *)

PARI

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

PARI

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

Python

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

Python

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

Formula

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