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

%I A065710 #44 Jul 12 2025 18:54:19
%S A065710 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 A065710 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 A065710 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 A065710 Number of 2's in the decimal expansion of 2^n.
%C A065710 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
%H A065710 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 A065710 a(n) = a(floor(n/10)) + [n == 2 (mod 10)], where [...] is the Iverson bracket. - _M. F. Hasler_, Feb 10 2023
%e A065710 2^31 = 2147483648 so a(31) = 1.
%p A065710 seq(numboccur(2, convert(2^n,base,10)),n=0..100); # _Robert Israel_, Jul 09 2025
%t A065710 Table[ Count[ IntegerDigits[2^n], 2], {n, 0, 100} ]
%o A065710 (PARI) a(n) = #select(x->(x==2), digits(2^n)); \\ _Michel Marcus_, Jun 15 2018
%o A065710 (Python)
%o A065710 def A065710(n):
%o A065710     return str(2**n).count('2') # _Chai Wah Wu_, Feb 14 2020
%Y A065710 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 A065710 Cf. A034293, A094776 (largest k for which 2^k has no digit n).
%K A065710 nonn,base
%O A065710 0,19
%A A065710 _Benoit Cloitre_, Dec 04 2001
%E A065710 More terms from _Robert G. Wilson v_, Dec 07 2001