A210434 Number of digits in 4^n.

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

Author

Luc Comeau-Montasse, Mar 21 2012

Keywords

• nonn
• base
• easy

Comments

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

Examples

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

Links

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

Crossrefs

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

Programs

Magma

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

Maple

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

Mathematica

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

PARI

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

PARI

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

Python

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

Formula

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