A129344 a(n) is the number of powers of 2 that have n decimal digits.

4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 3, 4, 3, 3

Offset

1,1

Comments

Ignoring the first term, first differences of A066343. - Andrew Woods, Jun 10 2013

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

For n>1, a(n) = floor(n*L)-floor((n-1)*L) where L = log(10)/log(2). - Andrew Woods, Jun 10 2013

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log_2(10) (A020862). - Amiram Eldar, Apr 09 2021

Example

a(1) is 4 because there are 4 one-digit powers of 2: 1, 2, 4, 8.

Mathematica

Table[Transpose[ Select[Table[{n, 2^n}, {n, 0, 310}], IntegerDigits[ #[[2]]][[1]] == 1 &]][[1]][[k]] - Transpose[ Select[Table[{n, 2^n}, {n, 0, 310}], IntegerDigits[ #[[2]]][[1]] == 1 &]][[1]][[k - 1]], {k, 2, 94}]
Join[{4}, Differences @ Table[Floor[n*Log2[10]], {n, 100}]] (* Amiram Eldar, Apr 09 2021 *)

Prog

(PARI) a(n) = my(k=0, i=0); while(#Str(2^k)!=n, k++); while(#Str(2^k)==n, i++; k++); i \\ Felix Fröhlich, Jan 19 2016
(Python)
def A129344(n): return -(m:=5**(n-1)).bit_length()+(5*m).bit_length()+1 if n>1 else 4 # Chai Wah Wu, Sep 08 2024

Crossrefs

First differences of A067497.

Cf. A000079, A020862, A066343.

Sequence in context: A026858 A188885 A362330 * A048853 A303701 A179845

Adjacent sequences: A129341 A129342 A129343 * A129345 A129346 A129347

Keyword

base,nonn

Author

Tanya Khovanova, May 28 2007

Status

approved