cp's OEIS Frontend

A217445 Numbers n such that n! has the same number of terminating zeros in bases 3 and 4.

%I A217445 #34 Jul 11 2020 14:11:32
%S A217445 1,2,4,5,6,7,10,11,12,13,14,18,19,21,22,23,33,36,37,38,42,43,46,47,51,
%T A217445 56,58,59,60,61,62,75,86,88,89,92,100,101,102,103,105,112,113,114,115,
%U A217445 120,121,122,124,125,138,139,141,147,153,159,164,166,167,168
%N A217445 Numbers n such that n! has the same number of terminating zeros in bases 3 and 4.
%C A217445 The number of zeros of n! base 3 is approaching n/2 as n grows. Similarly, the number of zeros of n! base 4 is approaching n/2 as n grows. Consequently, this sequence is expected to have high density.
%C A217445 From _Robert Israel_, Jan 19 2017: (Start)
%C A217445 Numbers n such that A000120(n) + (n + A000120(n) mod 2) = A053735(n).
%C A217445 Since typically A000120(n) ~ log_2(n) while typically A053735(n) ~ log_3(n), the density of this sequence should go to 0, contrary to the previous comment. (End)
%C A217445 Comment from _N. J. A. Sloane_, Dec 06 2019: (Start)
%C A217445 Appears to be the same as the list of positive numbers n such that the last nonzero digit of n! in base 12 belongs to the set [1, 2, 5, 7, 10, 11].
%C A217445 The first footnote in Deshouillers et al. (2016) says: "if the last nonzero digit of n! in base 12 belongs to {1, 2, 5, 7, 10, 11} then |(digit-sum of n in base 3) - (digit-sum of n in base 2)| is <= 1; this seems to occur infinitely many times." Compare A096288. (End)
%D A217445 Jean-Marc Deshouillers, Laurent Habsieger, Shanta Laishram, Bernard Landreau, Sums of the digits in bases 2 and 3, arXiv:1611.08180, 2016
%H A217445 Alois P. Heinz, <a href="/A217445/b217445.txt">Table of n, a(n) for n = 1..10000</a>
%e A217445 6! is 222200 in base 3 and 23100 in base 4, both of them have 2 zeros at the end, so 6 is in the sequence.
%p A217445 s2:= n -> convert(convert(n,base,2),`+`):
%p A217445 s3:= n -> convert(convert(n,base,3),`+`):
%p A217445 select(n -> s2(n) + (n+s2(n) mod 2) = s3(n), [$1..1000]); # _Robert Israel_, Jan 19 2017
%t A217445 sntzQ[n_]:=Module[{f=n!},Last[Split[IntegerDigits[f,3]]]==Last[ Split[ IntegerDigits[ f,4]]]]; Select[Range[200],sntzQ] (* _Harvey P. Dale_, Jul 11 2020 *)
%o A217445 (PARI) is(n)=my(L=log(n+1));sum(k=1,L\log(3),n\3^k)==sum(k=1,L\log(2),n>>k)\2 \\ _Charles R Greathouse IV_, Oct 04 2012
%Y A217445 Cf. A054861 (base 3), A090616 (base 4), A090622, A096288.
%K A217445 nonn,base
%O A217445 1,2
%A A217445 _Tanya Khovanova_, Oct 03 2012
%E A217445 More terms from _Alois P. Heinz_, Oct 03 2012