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A048656 a(n) is the number of unitary (and also of squarefree) divisors of n!. 31
1, 2, 4, 4, 8, 8, 16, 16, 16, 16, 32, 32, 64, 64, 64, 64, 128, 128, 256, 256, 256, 256, 512, 512, 512, 512, 512, 512, 1024, 1024, 2048, 2048, 2048, 2048, 2048, 2048, 4096, 4096, 4096, 4096, 8192, 8192, 16384, 16384, 16384, 16384, 32768, 32768, 32768, 32768 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Let K(n) be the field that is generated over the rationals Q by adjoining the square roots of the numbers 1,2,3,...,n, i.e., K(n) = Q(sqrt(1),sqrt(2),...,sqrt(n)); a(n) is the degree of this field over Q. - Avi Peretz (njk(AT)netvision.net.il), Mar 20 2001

For n>1, a(n) is the number of ways n! can be expressed as the product of two coprime integers p and q such that 0 < p/q < 1, if negative integers are considered as well. This is the answer to the 2nd problem of the International Mathematical Olympiad 2001. Example, for n = 3, the a(3) = 4 products are 3! = (-2)*(-3) = (-1)*(-6) = 1*6 = 2*3. - Bernard Schott, Jan 21 2021

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

International Mathematical Olympiad 2001, Hong Kong Preliminary Selection Contest, Problem 2.

Index to divisibility sequences.

Index to sequences related to Olympiads.

FORMULA

A001221(n!) = A000720(n) so a(n) = A034444(n!) = 2^A000720(n).

EXAMPLE

n=7, n! = 5040 = 16*9*5*7 with 4 distinct prime factors, so a(7) = A034444(7!) = 16.

PROG

(PARI) a(n)=2^primepi(n) \\ Charles R Greathouse IV, Apr 07 2012

CROSSREFS

Cf. A000720, A001221, A034444.

Sequence in context: A166632 A116596 A248692 * A107848 A285273 A188824

Adjacent sequences:  A048653 A048654 A048655 * A048657 A048658 A048659

KEYWORD

nonn

AUTHOR

Labos Elemer

STATUS

approved

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Last modified December 6 01:46 EST 2021. Contains 349558 sequences. (Running on oeis4.)