A048656 a(n) is the number of unitary (and also of squarefree) divisors of n!.

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

Offset: 1

Author

Labos Elemer

Keywords

• nonn

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

a(n) = number of subsets S of {1,2,...,n} such that every number in S is a prime. - Clark Kimberling, Sep 17 2022

Examples

			For n = 7, n! = 5040 = 16*9*5*7 with 4 distinct prime factors, so a(7) = A034444(7!) = 16.
The subsets S of {1, 2, 3, 4} such that every number in S is a prime are these: {}, {2}, {3}, {2, 3}; thus, a(4) = 4. - _Clark Kimberling_, Sep 17 2022
		

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.

Crossrefs

Cf. A000720, A001221, A034444, A357214, A357215.

Cf. A098882, A098990.

Programs

Mathematica

Table[2^PrimePi[n], {n, 1, 70}] (* Clark Kimberling, Sep 17 2022 *)

PARI

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

Formula

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

Sum_{n>=1} 1/a(n) = A098882 + 1 = A098990 - 1. - Amiram Eldar, Mar 13 2025