A034888 -id:A034888 - cp's OEIS frontend

A125122 First differences of A034888.

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

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Jan 10 2007

This sequence is not periodic because log(3)/log(10) is an irrational number. - T. D. Noe, Jan 10 2007

			a(1)=0 because 3^(1+1)=9 (one digit) 3^1=3 (one digit) and the difference is 0
a(4)=1 because 3^(4+1)=243 (three digits) 3^(4)=81 (two digits) and the difference is 1

P:=proc(n) local i,j,k,w,old; k:=3; 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);

Differences[IntegerLength[3^Range[0,110]]] (* Harvey P. Dale, Jan 28 2015 *)

a(n)=Number_of_digits{3^(n+1)}-Number_of digits{3^(n)} with n>=0.

A008564 Digits of powers of 3.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Irregular table with row length sequence A034888. - Jason Kimberley, Nov 26 2012

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

Muniru A Asiru, Table of n, a(n) for n = 0..5000
Kurt Mahler, On some irrational decimal fractions, Journal of Number Theory, Vol. 13, No. 2 (1981), pp. 268-269.

Cf. A000244, A034888.

Cf. A000455, A008565, A008566, A008567, A008568, A008569, A008570, A008571, A008572, A008573, A010054.

Flat(List([0..30],n->ListOfDigits(3^n))); # Muniru A Asiru, Sep 29 2018

Flatten[Table[IntegerDigits[3^n], {n, 0, 26}]] (* Alonso del Arte, Sep 29 2018 *)

A210434 Number of digits in 4^n.

Original entry on oeis.org

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

Offset: 0

Luc Comeau-Montasse, Mar 21 2012

Since log10(4) = A114493 ~ 0.60205 (= twice log10(2) = 0.30102999566...), the first 98 terms are equal to floor(n*3/5)+1. - M. F. Hasler, Mar 31 2025

			a(4) = 3 because 4^4 = 256, which has 3 digits.
a(5) = 4 because 4^5 = 1024, which has 4 digits.

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

Cf. A000302, A034887, A034888, A210062, A210435, A210436.

[#Intseq(4^n): n in [0..68]]; // Bruno Berselli, Mar 22 2012

a:= n-> length(4^n): seq(a(n), n=0..100); # Alois P. Heinz, Mar 22 2012

Table[Length[IntegerDigits[4^n]], {n, 0, 68}] (* Bruno Berselli, Mar 22 2012 *)

apply( {A210434(n)=logint(4^n,10)+1}, [0..66]) \\ M. F. Hasler, Mar 31 2025

a(n)=log(4)*n\log(10)+1 \\ correct up to n ~ 10^precision, with default precision = 38. - M. F. Hasler, Mar 31 2025

from math import log
def A210434(n): return int(n*log(4,10))+1 if n<1e16 else "not enough precision" # M. F. Hasler, Mar 31 2025

a(n) = A055642(A000302(n)) = A055642(4^n) = floor(log_10(10*(4^n))). - Jonathan Vos Post, Mar 22 2012

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]

A055255 Number of even digits in 3^n.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 1, 2, 2, 4, 4, 5, 4, 6, 6, 7, 6, 4, 5, 8, 9, 8, 6, 8, 8, 8, 6, 9, 6, 8, 6, 7, 8, 7, 8, 10, 9, 7, 9, 11, 11, 10, 11, 12, 10, 10, 8, 16, 13, 12, 11, 9, 16, 18, 14, 14, 17, 19, 16, 19, 13, 17, 14, 14, 14, 14, 14, 17, 16, 15, 16, 16, 15, 25, 23, 22

Offset: 0

Asher Auel, May 05 2000

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

Cf. A055256, A034888.

a(n) = #select(x->((x%2) == 0), digits(3^n)); \\ Michel Marcus, Nov 22 2019

A055256 Number of odd digits in 3^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 5, 4, 5, 3, 4, 3, 5, 3, 4, 3, 5, 7, 6, 4, 3, 5, 7, 6, 6, 7, 9, 7, 10, 9, 11, 11, 10, 12, 11, 10, 11, 14, 12, 10, 11, 12, 12, 11, 14, 14, 17, 9, 13, 14, 16, 18, 12, 10, 15, 15, 13, 11, 15, 12, 19, 15, 18, 19, 19, 20, 20, 18, 19, 21, 20, 21, 22

Offset: 0

Asher Auel, May 05 2000

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

Cf. A055255, A034888.

a(n) = #select(x->((x%2) == 1), digits(3^n)); \\ Michel Marcus, Nov 22 2019

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).

A385950 Number of digits in 3^(n!).

Original entry on oeis.org

1, 1, 1, 3, 12, 58, 344, 2405, 19238, 173138, 1731378, 19045154, 228541845, 2971043978, 41594615682, 623919235225, 9982707763599, 169706031981174, 3054708575661119, 58039462937561246, 1160789258751224920, 24376574433775723304, 536284637543065912676

Offset: 0

Vincenzo Librandi, Jul 27 2025

Cf. A055642, A100731.

Subsequence of A034888.

Mathematica
```
Array[Floor[#! Log10@3+1]&,22]
```

a(n) = A055642(A100731(n)).

cp's OEIS Frontend

A125122 First differences of A034888.

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A008564 Digits of powers of 3.

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A210434 Number of digits in 4^n.

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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

Maple

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A210062 Number of digits in 7^n.

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Magma

Mathematica

Formula

A055255 Number of even digits in 3^n.

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PARI