A210434 -id:A210434 - cp's OEIS frontend

A008565 Digits of powers of 4.

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

Offset: 0

N. J. A. Sloane

Irregular table with row length sequence A210434. - Jason Kimberley, Nov 26 2012

The constant whose decimal expansion is this sequence is irrational (Mahler, 1981). - Amiram Eldar, Mar 23 2025

			Triangle begins:
  1;
  4;
  1, 6;
  6, 4;
  2, 5, 6;
  1, 0, 2, 4;
  4, 0, 9, 6;
  1, 6, 3, 8, 4;
  6, 5, 5, 3, 6;
  2, 6, 2, 1, 4, 4;
  1, 0, 4, 8, 5, 7, 6;
  ...

Robert Israel, Table of n, a(n) for n = 0..10008 (rows 0 to 181, flattened)
Kurt Mahler, On some irrational decimal fractions, Journal of Number Theory, Vol. 13, No. 2 (1981), pp. 268-269.

Cf. A000302 (powers of 4), A210434.
Last elements of rows give A168428.
Cf. A000455, A008564, A008566, A008567, A008568, A008569, A008570, A008571, A008572, A008573, A010054.

R:= 1: t:= 1: count:= 1:
while count < 100 do
  t:= 4*t; L:= convert(t,base,10);
  count:= count+nops(L);
  R:= R, op(ListTools:-Reverse(L));
od:
R; # Robert Israel, May 05 2020

Table[IntegerDigits[4^i],{i,0,17}]//Flatten (* Stefano Spezia, Aug 06 2024 *)

A210435 Number of digits in 5^n.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 47

Offset: 0

Luc Comeau-Montasse, Mar 21 2012

			a(4) = 3 because 5^4 = 625, which has 3 digits.
a(5) = 4 because 5^5 = 3125, which has 4 digits.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Ana Rechtman, Octobre 2015, 3e Défi, Images des Mathématiques, CNRS, 2015 (in French).

Cf. A000351, A055642.

Number of digits in b^n: A034887 (b=2), A034888 (b=3), A210434 (b=4), A210435 (b=5), A210436 (b=6), A210062 (b=7).

[#Intseq(5^n): n in [0..67]]; // Bruno Berselli, Mar 22 2012

a:= n-> length(5^n): seq(a(n), n=0..100); # Alois P. Heinz, Mar 22 2012

Table[Length[IntegerDigits[5^n]], {n, 0, 67}] (* Bruno Berselli, Mar 22 2012 *)
IntegerLength[5^Range[0,70]] (* Harvey P. Dale, Mar 26 2013 *)

a(n) = #Str(5^n); \\ Michel Marcus, Oct 27 2015

a(n) = A055642(A000351(n)) = A055642(5^n) = floor(log_10(10*(5^n))). [Jonathan Vos Post, Mar 22 2012]

a(n) + A034887(n) = n+1. - Michel Marcus, Oct 27 2015

A210436 Number of digits in 6^n.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 5, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 21, 22, 22, 23, 24, 25, 25, 26, 27, 28, 29, 29, 30, 31, 32, 32, 33, 34, 35, 36, 36, 37, 38, 39, 39, 40, 41, 42, 43, 43, 44, 45, 46, 46, 47, 48, 49, 50, 50, 51, 52, 53

Offset: 0

Luc Comeau-Montasse, Mar 21 2012

			a(4) = 4 because 6^4 = 1296, which has 4 digits.
a(5) = 4 because 6^5 = 7776, which has 4 digits.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000400, A034887, A034888, A210062, A210434, A210435.

[#Intseq(6^n): n in [0..67]]; // Bruno Berselli, Mar 22 2012

a:= n-> length(6^n): seq (a(n), n=0..100); # Alois P. Heinz, Mar 22 2012

Table[Length[IntegerDigits[6^n]], {n, 0, 99}] (* Alonso del Arte, Mar 22 2012 *)

a(n) = A055642(A000400(n)) = A055642(6^n) = floor(log_10(10*(6^n))). - Jonathan Vos Post, Mar 23 2012

A210062 Number of digits in 7^n.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 11, 12, 13, 14, 15, 16, 17, 17, 18, 19, 20, 21, 22, 22, 23, 24, 25, 26, 27, 28, 28, 29, 30, 31, 32, 33, 33, 34, 35, 36, 37, 38, 39, 39, 40, 41, 42, 43, 44, 44, 45, 46, 47, 48, 49, 50, 50, 51, 52, 53, 54, 55, 55, 56, 57

Offset: 0

Luc Comeau-Montasse, Mar 16 2012

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000420, A055642.

Number of digits in b^n: A034887 (b=2), A034888 (b=3), A210434 (b=4), A210435 (b=5), A210436 (b=6), this sequence (b=7).

[#Intseq(7^n): n in [0..67]]; // Bruno Berselli, Mar 22 2012

Table[Length[IntegerDigits[7^n]], {n, 0, 100}] (* T. D. Noe, Mar 20 2012 *)

a(n) = A055642(A000420(n)) = A055642(7^n) = floor(log_10(10*(7^n))). [Jonathan Vos Post, Mar 23 2012]

A145875 Repdigit Kaprekar numbers.

Original entry on oeis.org

1, 9, 55, 99, 999, 7777, 9999, 22222, 99999, 999999, 4444444, 9999999, 88888888, 99999999, 999999999, 1111111111, 9999999999, 55555555555, 99999999999, 999999999999, 7777777777777, 9999999999999, 22222222222222, 99999999999999, 999999999999999, 4444444444444444, 9999999999999999, 88888888888888888, 99999999999999999

Offset: 1

Howard Berman (howard_berman(AT)hotmail.com), Oct 22 2008

Kaprekar numbers (A006886) all of whose digits are equal. - N. J. A. Sloane, Mar 26 2025
The only numbers where the repeated digit and the number of digits are the same are 1, 88888888 and 999999999.
Conjectures from Daniel Mondot, Mar 28 2025: (Start)
The sequence a(n)%10 (i.e. the last (or any) digit of a(n)), is 15-periodic. the sequence would be : 1,9,5,9,9,7,9,2,9,9,4,9,8,9,9, repeating.
For n>15, a(n) can be constructed from a(n-15) by concatenating to it 9 times a digit of a(n-15). (End)
The above two conjectures are linked, they are easily proved using modular arithmetic, and correspond to the explicit formula given below. - M. F. Hasler, Mar 28 2025

Daniel Mondot, Table of n, a(n) for n = 1..1000
Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315. See Sections 9.2.2 and 9.2.3.
Robert P. Munafo, Kaprekar sequences at MROB

Intersection of A006886 (Kaprekar numbers) and A010785 (repdigits).
A382161 is a subsequence.
A subsequence of A382163 (palindromic Kaprekar numbers).
Cf. A002275 (repunits), A210434 (#digits(4^n), equals #digits(a(n+1)) for n < 98 but not beyond, due to log10(4) ~ 0.6).

is_A145875(n)=is_A006886(n) && #Set(digits(n))==1 \\ M. F. Hasler, Mar 31 2025
apply( {A145875(n)=10^(n--*3\5+1)\9*if(bittest(5, n%5),[1,5,7,2,4,8][n%15*2\/5+1],9)}, [1..29]) \\ M. F. Hasler, Mar 28 2025

isk(k) = my(d=digits(k^2), nb=#d); if (nb%2, d=concat(0, d); nb++); fromdigits(Vec(d, nb/2)) + fromdigits(vector(nb/2, i, d[nb/2+i])) == k;
lista(nn) = my(list=List()); for (i=1, nn, for (d=1, 9, my(x = fromdigits(vector(i, k, d))); if (isk(x), listput(list, x)););); Vec(list); \\ Michel Marcus, Mar 29 2025

a(n) = d(n) * R(floor(n*3/5+2/5)), where R(n) = (10^n-1)/9 = A002275(n) and d = (1, 9, 5, 9, 9; 7, 9, 2, 9, 9; 4, 9, 8, 9, 9) repeating, where ";" is used just to emphasize the 3 similar subgroups of length 5, with 2nd, 4th and 5th element equal to 9. - M. F. Hasler, Mar 28 2025

Length (= number of digits) of the n-th term is floor((n+2)*3/5). - M. F. Hasler, Mar 31 2025

More terms from Gupta (2025) added by N. J. A. Sloane, Mar 26 2025

A348553 Number of digits in 11^n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 0

Seiichi Manyama, Oct 22 2021

			a(24) = 25 because 11^24 = 9849732675807611094711841, which has 25 digits.
a(25) = 27 because 11^25 = 108347059433883722041830251, which has 27 digits.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Number of digits in b^n: A034887 (b=2), A034888 (b=3), A210434 (b=4), A210435 (b=5), A210436 (b=6), A210062 (b=7), this sequence (b=11).

Cf. A001020, A055642.

a[n_] := IntegerLength[11^n]; Array[a, 100, 0] (* Amiram Eldar, Oct 22 2021 *)

PARI
```
a(n) = #Str(11^n);
```

def a(n): return len(str(11**n))
print([a(n) for n in range(98)]) # Michael S. Branicky, Oct 22 2021

a(n) = A055642(A001020(n)) = A055642(11^n).

A125144 Increments in the number of decimal digits of 4^n.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jan 11 2007

This sequence is not periodic because log(4)/log(10) is an irrational number. - T. D. Noe, Jan 25 2007

			a(1)=1 because 4^(1+1)=16 (two digits) 4^1=4 (one digit) and the difference is 1.
a(2)=0 because 4^(2+1)=64 (two digits) 4^(2)=16 (two digits) and the difference is 0.

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

Cf. A125117, A125122.

First differences of A210434.

P:=proc(n) local i,j,k,w,old; k:=4; for i from 1 by 1 to n do j:=k^i; w:=0; while j>0 do w:=w+1; j:=trunc(j/10); od; if i>1 then print(w-old); old:=w; else old:=w; fi; od; end: P(1000);
# alternative:
H:= [seq(ilog10(4^i),i=1..1001)]:
H[2..-1]-H[1..-2]; # Robert Israel, Jul 12 2018

Differences[IntegerLength[4^Range[100]]] (* Paolo Xausa, Jun 08 2024 *)

a(n) = #digits(4^(n+1)) - #digits(4^n); \\ Michel Marcus, Jul 12 2018

a(n)=Number_of_digits{4^(n+1)}-Number_of digits{4^(n)} with n>=0 and where "Number_of digits" is a hypothetical function giving the number of digits of the argument.

Offset corrected by Robert Israel, Jul 11 2018

cp's OEIS Frontend

A008565 Digits of powers of 4.

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A210435 Number of digits in 5^n.

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A210436 Number of digits in 6^n.

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Magma

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A210062 Number of digits in 7^n.

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Magma

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A145875 Repdigit Kaprekar numbers.

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A348553 Number of digits in 11^n.

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Mathematica

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A125144 Increments in the number of decimal digits of 4^n.

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