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A330024 a(n) = floor(n/z) where z is the number of zeros in the decimal expansion of 2^n, and a(n)=0 when z=0.

%I A330024 #42 Sep 08 2022 08:46:24
%S A330024 0,0,0,0,0,0,0,0,0,0,10,11,12,0,0,0,0,17,0,0,20,21,22,23,0,0,26,0,0,
%T A330024 29,30,0,0,0,0,0,0,0,38,0,40,41,21,14,44,45,46,47,48,0,50,0,26,17,27,
%U A330024 27,28,57,58,29,30,20,31,31,32,65,66,0,68,23,23,71,0
%N A330024 a(n) = floor(n/z) where z is the number of zeros in the decimal expansion of 2^n, and a(n)=0 when z=0.
%C A330024 Is a(229)=229 the largest term?
%C A330024 a(8949)=41; is 8949 the largest n such that a(n) >= 41?
%C A330024 Is 79391 the largest n such that a(n) <= 30?
%C A330024 Is 30 <= a(n) <= 36 true for all n >= 713789?
%C A330024 Conjecture: For every sequence which can be named as "digit k appears m times in the decimal expansion of 2^n", the sequences are finite for 0 <= k <= 9 and any given m >= 0. Every digit from 0 to 9 are inclined to appear an equal number of times in the decimal expansion of 2^n as n increases.
%H A330024 Metin Sariyar, <a href="/A330024/b330024.txt">Table of n, a(n) for n = 0..32000</a>
%F A330024 Conjecture: a(n) = 33 (= floor(10/log_10(2))) for all sufficiently large n. - _Pontus von Brömssen_, Jul 23 2021
%e A330024 a(11) = 11 because 2^11 = 2048, there is 1 zero in 2048 and the integer part of 11/1 is 11.
%p A330024 f:= proc(n) local z;
%p A330024   z:= numboccur(0,convert(2^n,base,10));
%p A330024   if z = 0 then 0 else floor(n/z) fi
%p A330024 end proc:
%p A330024 map(f, [$1..100]); # _Robert Israel_, Nov 28 2019
%t A330024 Do[z=DigitCount[2^n,10,0];an=IntegerPart[n/z];If[z==0,Print[0],Print[an]],{n,0,8000}]
%o A330024 (Magma) a:=[0]; for n in [1..72] do z:=Multiplicity(Intseq(2^n),0); if z ne 0 then  Append(~a,Floor(n/z)); else Append(~a,0); end if; end for; a; // _Marius A. Burtea_, Nov 27 2019
%o A330024 (PARI) a(n) = my(z=#select(d->!d, digits(2^n))); if (z, n\z, 0); \\ _Michel Marcus_, Jan 07 2020
%o A330024 (Python)
%o A330024 def A330024(n):
%o A330024   z=str(2**n).count('0')
%o A330024   return n//z if z else 0 # _Pontus von Brömssen_, Jul 24 2021
%Y A330024 Cf. A007377, A027870.
%K A330024 nonn,base
%O A330024 0,11
%A A330024 _Metin Sariyar_, Nov 27 2019