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A319316 Numbers k such that A090616(k) < A054861(k).

%I A319316 #25 Sep 19 2018 04:05:01
%S A319316 3,9,15,27,28,29,30,31,39,45,54,55,57,63,81,82,83,84,85,87,90,91,93,
%T A319316 94,95,99,108,109,110,111,117,118,119,123,126,127,135,162,163,165,171,
%U A319316 174,175,183,189,190,191,207,219,243,244,245,246,247,248,249,250,251
%N A319316 Numbers k such that A090616(k) < A054861(k).
%C A319316 Numbers k such that the highest power of 12 dividing n! is determined by the highest power of 4 dividing n!.
%C A319316 Note that A054861 and A090616 are both asymptotic to a(n) = n/2 + O(log(n)), nevertheless, it seems that the number of k such that A090616(k) is bigger predominates. Conjecture: the ratio of k <= N such that A090616(k) > A054861(k) tends to 1 as N tends to infinity, while the ratio of k <= N such that A090616(k) < A054861(k) and A090616(k) = A054861(k) both tend to 0.
%C A319316 Number of k in range [0, N] such that A090616(k) =, < or > A054861(k):
%C A319316 ..N....A090616(k) = A054861(k)...A090616(k) < A054861(k)...A090616(k) > A054861(k)
%C A319316 10^2...............38........................26........................37
%C A319316 10^3..............344.......................228.......................429
%C A319316 10^4.............2703......................2227......................5071
%C A319316 10^5............23003.....................19892.....................57106
%C A319316 10^6...........203478....................185152....................611371
%C A319316 10^7..........1762288...................1726062...................6511651
%H A319316 Jianing Song, <a href="/A319316/b319316.txt">Table of n, a(n) for n = 1..2227</a> (all terms <= 10000)
%e A319316 The highest power of 3 dividing 9! is 3^4, while the highest power of 4 dividing 9! is 4^3, so 9 is a term, and the highest power of 12 dividing 9! is 12^3.
%e A319316 The highest power of 3 dividing 15! is 3^6, while the highest power of 4 dividing 15! is 4^5, so 15 is a term, and the highest power of 12 dividing 15! is 12^5.
%o A319316 (PARI) isA319316(n)=(n-vecsum(digits(n, 2)))\2<(n-vecsum(digits(n, 3)))\2
%Y A319316 Cf. A217445 (k such that A090616(k) = A054861(k)), A319317 (k such that A090616(k) > A054861(k)).
%K A319316 nonn
%O A319316 1,1
%A A319316 _Jianing Song_, Sep 17 2018