A232744 Numbers k for which the largest m such that m! divides k is odd.

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, 41, 42, 43, 45, 47, 49, 51, 53, 54, 55, 57, 59, 60, 61, 63, 65, 66, 67, 69, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 97, 99, 101, 102, 103, 105

Offset: 1

Author

Antti Karttunen, Dec 01 2013

Keywords

• nonn

Comments

Numbers k for which A055881(k) is odd.

Equally: Numbers k which have an even number of the trailing zeros in their factorial base representation A007623(k).

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.

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

Links

• Antti Karttunen, Table of n, a(n) for n = 1..9558

Crossrefs

Complement: A232745. Cf. also A055881, A007623, A232741-A232743.

Analogous sequences for binary system: A003159 & A036554.

Cf. A002467, A068996.

Programs

Mathematica

seq[max_] := Select[Range[max!], EvenQ @ LengthWhile[Reverse @ IntegerDigits[#, MixedRadix[Range[max, 2, -1]]], #1 == 0 &] &]; seq[5] (* Amiram Eldar, Feb 26 2021 *)

Formula

a(1)=1, and for n>1, a(n) = a(n-1) + (2 - A000035(A055881(a(n-1)+1))).