%I #16 Feb 26 2021 20:12:08
%S 1,3,5,6,7,9,11,12,13,15,17,18,19,21,23,25,27,29,30,31,33,35,36,37,39,
%T 41,42,43,45,47,49,51,53,54,55,57,59,60,61,63,65,66,67,69,71,73,75,77,
%U 78,79,81,83,84,85,87,89,90,91,93,95,97,99,101,102,103,105
%N Numbers k for which the largest m such that m! divides k is odd.
%C Numbers k for which A055881(k) is odd.
%C Equally: Numbers k which have an even 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 1!, except that it excludes from those the multiples of 2!, except that it includes the multiples of 3! (6), except that it excludes the multiples of 4! (24), except that it includes the multiples of 5! (120), except that it excludes the multiples of 6! (720), except that it includes the multiples of 7! (5040), except that it excludes the multiples of 8! (40320), except that it includes the multiples of 9! (362880), and so on, ad infinitum.
%C The number of terms not exceeding m! for m>=1 is A002467(m). The asymptotic density of this sequence is 1 - 1/e (A068996). - _Amiram Eldar_, Feb 26 2021
%H Antti Karttunen, <a href="/A232744/b232744.txt">Table of n, a(n) for n = 1..9558</a>
%F a(1)=1, and for n>1, a(n) = a(n-1) + (2 - A000035(A055881(a(n-1)+1))).
%t seq[max_] := Select[Range[max!], EvenQ @ 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 A232744 (MATCHING-POS 1 1 (lambda (n) (odd? (A055881 n)))))
%o ;; Alternative implementation with a simple recurrence:
%o (definec (A232744 n) (if (<= n 1) n (+ (A232744 (- n 1)) (- 2 (A000035 (A055881 (+ 1 (A232744 (- n 1)))))))))
%Y Complement: A232745. Cf. also A055881, A007623, A232741-A232743.
%Y Analogous sequences for binary system: A003159 & A036554.
%Y Cf. A002467, A068996.
%K nonn
%O 1,2
%A _Antti Karttunen_, Dec 01 2013