A004216 -id:A004216 - 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

A020338 Doublets: base-10 representation is the juxtaposition of two identical strings.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 1010, 1111, 1212, 1313, 1414, 1515, 1616, 1717, 1818, 1919, 2020, 2121, 2222, 2323, 2424, 2525, 2626, 2727, 2828, 2929, 3030, 3131, 3232, 3333, 3434, 3535, 3636, 3737, 3838, 3939, 4040, 4141, 4242, 4343, 4444, 4545, 4646

Offset: 1

David W. Wilson

Reinhard Zumkeller, Table of n, a(n) for n = 1..9999
Eric Weisstein's World of Mathematics, Concatenation

Cf. concatenation of n and k*n: this sequence (k=1), A019550 (k=2), A019551 (k=3), A019552 (k=4), A019553 (k=5), A009440 (k=6), A009441 (k=7), A009470 (k=8), A009474 (k=9).

Cf. A004216, A056524.

Flat(List([1..2],d->List([10^(d-1)..10^d-1],n->(10^d+1)*n))); # Muniru A Asiru, Mar 31 2018

a020338 n = read (ns ++ ns) :: Integer  where ns = show n
-- Reinhard Zumkeller, Jun 07 2015

[Seqint(Intseq(n) cat Intseq(n)): n in [1..46]]; // Bruno Berselli, Mar 20 2013

seq(seq((10^d+1)*n, n = 10^(d-1)..10^d-1),d=1..3); # Robert Israel, Jan 02 2015

nxt[n_]:=Module[{idn=IntegerDigits[n], idn1=IntegerDigits[n]}, FromDigits[Join[idn, idn1]]];Array[nxt, 100] (* Vincenzo Librandi, Feb 04 2014 *)

a(n) = eval(Str(n,n)); \\ Michel Marcus, Sep 10 2015

[int(str(n)+str(n)) for n in range(1,47)] # Danny Rorabaugh, Oct 10 2015

a(n) = n*10^(A004216(n)+1) + n. - Reinhard Zumkeller, Aug 11 2007
G.f.: 11*x/(1-x)^2 - Sum_{d >= 1} 9*x^(10^d)*(100^d*x-10^d*x-100^d)/(1-x)^2. - Robert Israel, Jan 02 2015
a(n) = n || n. (Where "||" denotes "concatenate". See link "Concatenation".) - Halfdan Skjerning, Apr 01 2018

A000461 Concatenate n n times.

Original entry on oeis.org

1, 22, 333, 4444, 55555, 666666, 7777777, 88888888, 999999999, 10101010101010101010, 1111111111111111111111, 121212121212121212121212, 13131313131313131313131313, 1414141414141414141414141414, 151515151515151515151515151515, 16161616161616161616161616161616

Offset: 1

John Radu (Suttones(AT)aol.com)

			From _Bruno Berselli_, Oct 05 2018: (Start)
.         1 * 9 = 09
.        22 * 9 = 198
.       333 * 9 = 2997
.      4444 * 9 = 39996
.     55555 * 9 = 499995
.    666666 * 9 = 5999994
.   7777777 * 9 = 69999993
.  88888888 * 9 = 799999992
. 999999999 * 9 = 8999999991
(End)

F. Smarandache, "Properties of the numbers", Univ. of Craiova Archives, 1975; Arizona State University Special Collections, Tempe, AZ.

Reinhard Zumkeller, Table of n, a(n) for n = 1..333
Eric Weisstein's World of Mathematics, Smarandache Sequences

Cf. A048376, A053422.

a000461 n = (read $ concat $ replicate n $ show n) :: Integer
-- Reinhard Zumkeller, Apr 26 2011

a:= n-> parse(cat(n$n)):
seq(a(n), n=1..20);  # Alois P. Heinz, Apr 26 2011

Table[Sum[(n)*10^(i*(Floor[Log[10, n]] + 1)), {i, 0, n - 1}], {n, 1, 30}] (* José de Jesús Camacho Medina, Dec 10 2014 *)
Table[FromDigits[Flatten[IntegerDigits/@Table[n,{n}]]],{n,15}] (* Harvey P. Dale, Mar 01 2015 *)
Table[FromDigits[PadRight[{},n IntegerLength[n],IntegerDigits[n]]],{n,15}] (* Harvey P. Dale, Jun 19 2016 *)

a(n) = eval(concat(apply(x->Str(x), vector(n, k, n)))); \\ Michel Marcus, Oct 05 2018; Feb 12 2023

def a(n): return int(str(n)*n)
print([a(n) for n in range(1, 17)]) # Michael S. Branicky, Jan 22 2021

a(n) = n*(10^(n*L(n))-1)/(10^L(n)-1) where L(n) = A004216(n)+1 = floor(log_10(10n)). - Henry Bottomley, Jun 01 2000
A055642(a(n)) = n * A055642(n). - Reinhard Zumkeller, Apr 26 2011
a(n) = Sum_{i=0..n-1} (n*10^(i*(floor(log(10, n)) + 1))). - José de Jesús Camacho Medina, Dec 10 2014

A045532 Concatenate n with n-th prime.

Original entry on oeis.org

12, 23, 35, 47, 511, 613, 717, 819, 923, 1029, 1131, 1237, 1341, 1443, 1547, 1653, 1759, 1861, 1967, 2071, 2173, 2279, 2383, 2489, 2597, 26101, 27103, 28107, 29109, 30113, 31127, 32131, 33137, 34139, 35149, 36151, 37157, 38163, 39167, 40173, 41179, 42181

Offset: 1

Jeff Burch

Triangular numbers are 1653, 32131, 79401, ... - Ali Adams, Feb 04 2020

The next such terms are 173340627863131 and 1454987833022905581. - Giovanni Resta, Feb 04 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A075110.

Cf. A085451. [Reinhard Zumkeller, Jun 30 2010]

a045532 n = read $ show n ++ show (a000040 n) :: Integer
-- Reinhard Zumkeller, Jul 08 2014

[Seqint(Intseq(NthPrime(n)) cat Intseq(n)): n in [1..45]]; // Vincenzo Librandi, Apr 13 2019

Table[FromDigits[Join[IntegerDigits[n], IntegerDigits[Prime[n]]]], {n, 40}] (* Vincenzo Librandi, Apr 13 2019 *)
#[[1]]*10^IntegerLength[#[[2]]]+#[[2]]&/@Table[{n,Prime[n]},{n,50}] (* Harvey P. Dale, Oct 11 2024 *)

a(n) = eval(Str(n,prime(n))); \\ Michel Marcus, Apr 13 2019, simplified by M. F. Hasler, Feb 05 2020

from sympy import prime
def a(n): return int(str(n) + str(prime(n)))
print([a(n) for n in range(1, 43)]) # Michael S. Branicky, Dec 23 2021

a(n) = n*10^(A004216(A000040(n))+1) + A000040(n). - Reinhard Zumkeller, Sep 03 2002

A075110 Concatenation of n-th prime and n in decimal notation.

Original entry on oeis.org

21, 32, 53, 74, 115, 136, 177, 198, 239, 2910, 3111, 3712, 4113, 4314, 4715, 5316, 5917, 6118, 6719, 7120, 7321, 7922, 8323, 8924, 9725, 10126, 10327, 10728, 10929, 11330, 12731, 13132, 13733, 13934, 14935, 15136, 15737, 16338, 16739, 17340

Offset: 1

Reinhard Zumkeller, Sep 03 2002

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A045532.

a075110 n = read $ show (a000040 n) ++ show n :: Integer
-- Reinhard Zumkeller, Jul 08 2014

[Seqint(Intseq(n) cat Intseq(NthPrime(n))): n in [1..46]]; // Vincenzo Librandi, Mar 24 2019

Table[FromDigits[Join[IntegerDigits[Prime[n]], IntegerDigits[n]]], {n, 40}] (* Vincenzo Librandi, Mar 24 2019 *)

a(n) = eval(Str(prime(n), n)); \\ Michel Marcus, Mar 24 2019

a(n) = A000040(n)*10^(A004216(n)+1) + n.

A004218 a(n) = log_10(n) rounded up.

Original entry on oeis.org

0, 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

Offset: 1

N. J. A. Sloane

nonn

a(n) is the number of terms in the sequence A011557 (Powers of 10) that are less than n. For n > 1, a(n) is the number of digits in n-1. - Tanya Khovanova, Jun 22 2007

			From _M. F. Hasler_, Dec 07 2018: (Start)
log_10(1) = 0, therefore a(1) = 0.
log_10(2) = 0.301..., therefore a(2) = 1.
log_10(9) = 0.954..., therefore a(9) = 1.
log_10(10) = 1, therefore a(10) = 1.
log_10(11) = 1.04..., therefore a(11) = 2.
log_10(99) = 1.9956..., therefore a(99) = 2.
log_10(100) = 2, therefore a(100) = 2.
log_10(101) = 2.004..., therefore a(101) = 3. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A004216, A007953, A055642.

a004218 n = if n == 1 then 0 else 1 + a004216 (n - 1)

A004218 := proc(n)
    ceil(log[10](n)) ;
end proc:
seq(A004218(n),n=1..120) ; # R. J. Mathar, May 16 2023

Array[Ceiling[Log10[#]] &, 100] (* Amiram Eldar, Dec 08 2018 *)

A004218(n)=logint(n-(n>1),10)+1 \\ M. F. Hasler, Dec 07 2018

a(1) = 0, a(n) = 1 + A004216(n-1) for n > 1. - Reinhard Zumkeller, Dec 22 2012

a(n) = A055642(n-1) for all n > 1. a(n+1) is the number of decimal digits of n if 0 is considered to have 0 digits. - M. F. Hasler, Dec 07 2018

A211666 Number of iterations log_10(log_10(log_10(...(n)...))) such that the result is < 2.

Original entry on oeis.org

0, 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, 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, 2, 2, 2, 2, 2, 2

Offset: 1

Hieronymus Fischer, Apr 30 2012

Different from A004216, A057427 and A185114.
For a general definition like "Number of iterations log_p(log_p(log_p(...(n)...))) such that the result is < q", where p > 1, q > 0, the resulting g.f. is
g(x) = (1/(1-x))*Sum_{k>=1} x^(E_{i=1..k} b(i,k)), where b(i,k)=p for i

			a(n) = 0, 1, 2, 3 for n = 1, 2, 10^2, 10^10^2 (= 1, 2, 100, 10^100).

Cf. A001069, A010096, A211662, A211664, A211668, A211670.

With the exponentiation definition E_{i=1..n} c(i) := c(1)^(c(2)^(c(3)^(...(c(n-1)^(c(n)))...))); E_{i=1..0} c := 1; example: E_{i=1..3} 10 = 10^(10^10) = 10^10000000000, we get:
a(E_{i=1..n} 10) = a(E_{i=1..n-1} 10)+1, for n>=1.
G.f.: g(x) = (1/(1-x))*Sum_{k>=1} x^(E_{i=1..k} b(i,k)), where b(i,k)=10 for iThe explicit first terms of the g.f. are g(x) = (x^2+x^100+x^(10^100)+...)/(1-x).

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

A097413 Initial decimal digit of n^9.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Aug 16 2004

			1, 512, 19683, 262144, 1953125, 10077696, 40353607, 134217728, 387420489, 1000000000, ...

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Gelfand's Question

Cf. A000027, A000030, A002993, A002994, A004216, A097408, A097409, A097410, A097411, A097412, A097414.

idd:= n -> floor(n/10^ilog10(n)):
seq(idd(n^9),n=2..100); # Robert Israel, Jan 28 2016

Table[IntegerDigits[n^9][[1]],{n,120}] (* Harvey P. Dale, Aug 25 2023 *)

a(n) = A000030(n^9) = floor(n^9/10^A004216(n^9)). - Robert Israel, Jan 28 2016

A064222 a(0) = 0; a(n) = DecimalDigitsSortedDecreasing(a(n-1) + 1) for n > 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 22, 32, 33, 43, 44, 54, 55, 65, 66, 76, 77, 87, 88, 98, 99, 100, 110, 111, 211, 221, 222, 322, 332, 333, 433, 443, 444, 544, 554, 555, 655, 665, 666, 766, 776, 777, 877, 887, 888, 988, 998, 999, 1000, 1100, 1110, 1111, 2111

Offset: 0

Reinhard Zumkeller, Sep 21 2001

a(n) = A004186(a(n-1) + 1). - Reinhard Zumkeller, Oct 31 2007

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A004186, A004216, A027468.

a064222 n = a064222_list !! n
a064222_list = iterate (a004186 . (+ 1)) 0
-- Reinhard Zumkeller, Apr 11 2012

NestList[FromDigits[Sort[IntegerDigits[#+1],Greater]]&,0,60] (* Harvey P. Dale, Sep 04 2011 *)

a(n+1) = (d+0^d)*10^floor(log_10(a(n)+1)) + (1-0^d)*floor(a(n)/10), where d = (a(n)+1) mod 10. - Reinhard Zumkeller, Oct 31 2007

a(n) = (ceiling( (n-G(D(n)-1))/D(n) )*(10^D(n) -1) - 10^( (G(D(n)-1)-n) mod (D(n)) ) + 1)/9, for n>0, where D(n) = floor( (sqrt(8n+1)+3)/6 ) is the number of digits in a(n), and G(k) = A027468(k) = 9*k*(k+1)/2. - Stefan Alexandru Avram, May 24 2023

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cp's OEIS Frontend

A055642 Number of digits in the decimal expansion of n.

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A020338 Doublets: base-10 representation is the juxtaposition of two identical strings.

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A000461 Concatenate n n times.

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A045532 Concatenate n with n-th prime.

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A075110 Concatenation of n-th prime and n in decimal notation.

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A004218 a(n) = log_10(n) rounded up.

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