%I #13 Feb 26 2021 20:12:14
%S 2,4,8,10,14,16,20,22,24,26,28,32,34,38,40,44,46,48,50,52,56,58,62,64,
%T 68,70,72,74,76,80,82,86,88,92,94,96,98,100,104,106,110,112,116,118,
%U 122,124,128,130,134,136,140,142,144,146,148,152,154,158,160,164
%N Numbers k for which the largest m such that m! divides k is even.
%C Numbers k for which A055881(k) is even.
%C Equally: Numbers k which have an odd number of the trailing zeros in their factorial base representation A007623(k).
%C The sequence can be described in the following manner: Sequence includes all multiples of 2! (even numbers), except that it excludes from those the multiples of 3! (6), except that it includes the multiples of 4! (24), except that it excludes the multiples of 5! (120), except that it includes the multiples of 6! (720), except that it excludes the multiples of 7! (5040), except that it includes the multiples of 8! (40320), except that it excludes the multiples of 9! (362880), except that it includes the multiples of 10! (3628800), except that ..., ad infinitum.
%C The number of terms not exceeding m! for m>=1 is A000166(m). The asymptotic density of this sequence is 1/e (A068985). - _Amiram Eldar_, Feb 26 2021
%H Antti Karttunen, <a href="/A232745/b232745.txt">Table of n, a(n) for n = 1..14833</a>
%t seq[max_] := Select[Range[max!], OddQ @ LengthWhile[Reverse @ IntegerDigits[#, MixedRadix[Range[max, 2, -1]]], #1 == 0 &] &]; seq[5] (* _Amiram Eldar_, Feb 26 2021 *)
%o (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o (define A232745 (MATCHING-POS 1 1 (lambda (n) (even? (A055881 n)))))
%Y Complement: A232744.
%Y b(n) = A153880(A232744(n)) gives a subset of this sequence.
%Y Cf. also A055881, A007623, A232741-A232743.
%Y Analogous sequences for binary system: A003159 & A036554.
%Y Cf. A000166, A068985.
%K nonn
%O 1,1
%A _Antti Karttunen_, Dec 01 2013