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

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

See A034293 for indices of zeros: It is conjectured that the last 0 appears at index 168 = A094776(2). More generally, I conjecture that the last occurrence of the term x = 0, 1, 2, 3, ... is at index i = (168, 176, 186, 268, 423, 361, 472, 555, 470, 562, 563, 735, ...). - M. F. Hasler, Feb 10 2023

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

Mathematica

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

Prog

(PARI) Count(x, d)= { local(c=0, f); while (x>9, f=x-10*(x\10); if (f==d, c++); x\=10); if (x==d, c++); return(c) }
{ for (n=0, 1000, a=Count(2^n, 2); write("b065710.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 26 2009
(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.

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