A065710 Number of 2's in the decimal expansion of 2^n.

0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 2, 2, 0, 2, 0, 0, 1, 1, 0, 2, 1, 1, 1, 1, 2, 1, 0, 0, 0, 1, 1, 0, 1, 4, 0, 3, 1, 2, 0, 1, 1, 3, 3, 3, 1, 2, 0, 1, 2, 1, 2, 2, 2, 3, 1, 3, 0, 2, 2, 3, 3, 2, 2, 4, 4, 4, 0, 1, 2, 4, 3, 1, 3, 6, 2, 0, 2, 4, 4, 4, 2, 3, 6, 2, 1, 5, 1, 2, 4, 4, 1, 2, 6

Offset: 0

Author

Benoit Cloitre, Dec 04 2001

Keywords

• nonn
• base

Comments

See A034293 for indices of zeros: It is conjectured that the last 0 appears at index 168 = A094776(2). More generally, I conjecture that any value x = 0, 1, 2, 3, ... occurs only a finite number of times N(x) = 23, 35, 28, 26, 41, 37, 34, 26, 34, 38, 33, 41, ... in this sequence, for the last time at a well defined index i(x) = 168, 176, 186, 268, 423, 361, 472, 555, 470, 562, 563, 735, .... - M. F. Hasler, Feb 10 2023, edited by M. F. Hasler, Jul 09 2025

Examples

			2^31 = 2147483648 so a(31) = 1.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 0..10000 (first 1001 terms from Harry J. Smith), Feb 10 2023

Crossrefs

Cf. 0's A027870, 1's A065712, 3's A065714, 4's A065715, 5's A065716, 6's A065717, 7's A065718, 8's A065719, 9's A065744.

Cf. A034293, A094776 (largest k for which 2^k has no digit n).

Programs

Maple

seq(numboccur(2, convert(2^n,base,10)),n=0..100); # Robert Israel, Jul 09 2025

Mathematica

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

PARI

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

Python

def A065710(n):
    return str(2**n).count('2') # Chai Wah Wu, Feb 14 2020

Formula

a(n) = a(floor(n/10)) + [n == 2 (mod 10)], where [...] is the Iverson bracket. - M. F. Hasler, Feb 10 2023

Extensions

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