A232744 - OEIS

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

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OFFSET

1,2

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

FORMULA

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

MATHEMATICA

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

PROG

(Scheme, with Antti Karttunen's IntSeq-library)

(define A232744 (MATCHING-POS 1 1 (lambda (n) (odd? (A055881 n)))))

;; Alternative implementation with a simple recurrence:

(definec (A232744 n) (if (<= n 1) n (+ (A232744 (- n 1)) (- 2 (A000035 (A055881 (+ 1 (A232744 (- n 1)))))))))

CROSSREFS

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

Analogous sequences for binary system: A003159 & A036554.

Cf. A002467, A068996.

Sequence in context: A301729 A047273 A342051 * A284015 A286908 A284852

Adjacent sequences: A232741 A232742 A232743 * A232745 A232746 A232747

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 01 2013

STATUS

approved