A065710 - OEIS

%I #33 Feb 11 2023 07:46:02

%S 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,

%T 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,

%U 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

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

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

%C 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

%H M. F. Hasler, <a href="/A065710/b065710.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from Harry J. Smith), Feb 10 2023

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

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

%o (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) }

%o { for (n=0, 1000, a=Count(2^n, 2); write("b065710.txt", n, " ", a) ) } \\ _Harry J. Smith_, Oct 26 2009

%o (PARI) a(n) = #select(x->(x==2), digits(2^n)); \\ _Michel Marcus_, Jun 15 2018

%o (Python)

%o def A065710(n):

%o     return str(2**n).count('2') # _Chai Wah Wu_, Feb 14 2020

%Y 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.

%Y Cf. A034293, A094776.

%K nonn,base

%O 0,19

%A _Benoit Cloitre_, Dec 04 2001

%E More terms from _Robert G. Wilson v_, Dec 07 2001