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

0,19

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

Links

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

Formula

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

Example

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

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} ]

Prog

(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

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).

Sequence in context: A244606 A273127 A103272 * A089799 A073464 A142242

Adjacent sequences: A065707 A065708 A065709 * A065711 A065712 A065713

Keyword

nonn,base

Author

Benoit Cloitre, Dec 04 2001

Extensions

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

Status

approved