A065714 Number of 3's in decimal expansion of 2^n.

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 3, 0, 1, 1, 1, 1, 1, 0, 1, 0, 4, 1, 3, 0, 1, 0, 1, 1, 1, 0, 3, 1, 3, 0, 2, 0, 1, 2, 0, 1, 2, 2, 0, 2, 3, 0, 4, 1, 3, 1, 4, 2, 1, 1, 1, 2, 3, 2, 3, 1, 2, 4, 1, 4, 3, 0, 3, 2, 3, 4, 4, 3, 3, 2, 1, 3, 0, 0, 4, 2, 2, 6, 1, 4, 4, 2

Offset: 0

Author

Benoit Cloitre, Dec 04 2001

Keywords

• nonn
• base

Comments

I conjecture that any value x = 0, 1, 2, ... occurs only a finite number of times N(x) = 34, 34, 24, 34, 39, 34, 35, 34, 35, 32, 33, 31, ... in this sequence, for the last time at well defined indices i(x) = 153, 139, 226, 237, 308, 386, 413, 506, 461, 578, 644, 732, 857, 657, 743, 768, 784, 848, 906, ... - M. F. Hasler, Jul 09 2025

Examples

			2^5 = 32 so a(5)=1.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000079 (powers of 2), A035058 (2^n does not contain the digit 3).

Similar for other digits: A027870 (0's), A065712 (1's), A065710 (2's), this (3's), A065715 (4's), A065716 (5's), A065717 (6's), A065718 (7's), A065719 (8's), A065744 (9's).

Cf. A094776 (index of last occurrence of digit n in powers of 2).

Programs

Mathematica

Table[ Count[ IntegerDigits[2^n], 3], {n, 0, 100} ]

PARI

a(n) = #select(x->(x==3), digits(2^n)); \\ Michel Marcus, Jun 15 2018

Python

def A065714(n):
    return str(2**n).count('3') # Chai Wah Wu, Feb 14 2020

Extensions

More terms from Robert G. Wilson v, Dec 07 2001