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A319317 Numbers k such that A090616(k) > A054861(k).

%I A319317 #23 Nov 13 2021 04:41:17
%S A319317 8,16,17,20,24,25,26,32,34,35,40,41,44,48,49,50,52,53,64,65,66,67,68,
%T A319317 69,70,71,72,73,74,76,77,78,79,80,96,97,98,104,106,107,116,128,129,
%U A319317 130,131,132,133,134,136,137,140,142,143,144,145,146,148,149,150,151
%N A319317 Numbers k such that A090616(k) > A054861(k).
%C A319317 Numbers k such that the highest power of 12 dividing n! is determined by the highest power of 3 dividing n!.
%C A319317 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 thatA090616(k) < A054861(k) and A090616(k) = A054861(k) both tend to 0.
%C A319317 Number of k in range [0, N] such that A090616(k) =, < or > A054861(k):
%C A319317   N    A090616(k) = A054861(k)   A090616(k) < A054861(k)   A090616(k) > A054861(k)
%C A319317 10^2               38                        26                        37
%C A319317 10^3              344                       228                       429
%C A319317 10^4             2703                      2227                      5071
%C A319317 10^5            23003                     19892                     57106
%C A319317 10^6           203478                    185152                    611371
%C A319317 10^7          1762288                   1726062                   6511651
%H A319317 Jianing Song, <a href="/A319317/b319317.txt">Table of n, a(n) for n = 1..5071</a> (all terms <= 10000)
%e A319317 The highest power of 3 dividing 8! is 3^2, while the highest power of 4 dividing 8! is 4^3, so 8 is a term, and the highest power of 12 dividing 8! is 12^2.
%e A319317 The highest power of 3 dividing 16! is 3^6, while the highest power of 4 dividing 16! is 4^7, so 16 is a term, and the highest power of 12 dividing 16! is 12^6.
%o A319317 (PARI) isA319317(n)=(n-vecsum(digits(n, 2)))\2>(n-vecsum(digits(n, 3)))\2
%Y A319317 Cf. A217445 (k such that A090616(k) = A054861(k)), A319316 (k such that A090616(k) < A054861(k)).
%K A319317 nonn
%O A319317 1,1
%A A319317 _Jianing Song_, Sep 17 2018