A090619 Highest power of 12 dividing n!.

0, 0, 0, 0, 1, 1, 2, 2, 2, 3, 4, 4, 5, 5, 5, 5, 6, 6, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 15, 17, 17, 17, 17, 18, 18, 19, 19, 19, 20, 21, 21, 22, 22, 22, 23, 23, 23, 25, 25, 26, 26, 27, 27, 28, 28, 28, 28, 30, 30, 31, 31, 31, 32, 32, 32, 34, 34, 34, 35, 35

Offset

0,7

Comments

Most sequences of the form "highest power of k dividing n!" essentially depend on one of the primes or prime powers dividing k. But in this case, the sequences with k=3 (A054861) and k=4 (A090616) are both close to n/2 and vary in which one is lower for different values of n.

a(2^n) = A090616(2^n) and a(3^n-1) = A090616(3^n-1) while a(2^n-1) = A054861(2^n-1) and a(3^n) = A054861(3^n). - Robert Israel, Mar 25 2018

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Formula

a(n) =A090622(n, 12) =min(A054861(n), A090616(n)). Close to n/2, indeed for n>3: n/2-log3(n+1) <= a(n) < n/2.

Example

a(6)=2 since 6!=720=12^2*5.

Maple

f2:= n -> n - convert(convert(n, base, 2), `+`):
f3:= n -> (n - convert(convert(n, base, 3), `+`))/2:
f:= n -> min(f3(n), floor(f2(n)/2)):
f(0):= 0:
map(f, [$0..100]); # Robert Israel, Mar 23 2018

Mathematica

Table[IntegerExponent[n!, 12], {n, 0, 100}] (* Jean-François Alcover, Mar 26 2018 *)

Prog

(PARI) a(n) = valuation(n!, 12); \\ Michel Marcus, Mar 24 2018

Crossrefs

Cf. A011371, A054861, A090616, A027868, A054896, A090617, A090618.

Sequence in context: A365720 A332251 A342249 * A249038 A300076 A029113

Adjacent sequences: A090616 A090617 A090618 * A090620 A090621 A090622

Keyword

nonn

Author

Henry Bottomley, Dec 06 2003

Status

approved