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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. 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 STATUS approved

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