A160375 Given n, let S denote the set of numbers c_1*c_2*...*c_n where 1<=c_1<=c_2<=...<=c_n<=n; a(n) = number of members of S that have a unique representation of this form.

1, 3, 10, 16, 61, 81, 337, 477, 601, 901, 4291, 5798, 27314, 33671, 45732, 59397, 299745, 421363, 2090647, 2739022, 4597263, 5401826, 27510715, 23666955

Offset

1,2

Comments

Number of combinations as in A001700.

From David A. Corneth, Sep 26 2016: (Start)

a(n + 1) / a(n) is fairly large if n + 1 is prime; for the given data, it's at least three. In the other cases it's less than 2.

Let p be a distinct product as described in the name. We look at the factors rather than the result. For n = 4, we see the product p = 1*2*3*3.

Let F(p) be a vector of size n which counts the frequency F_e of each e where 1 <= e <= n. For n = 4 and the product we find (1,1,2,0).

For n = 6, we can put the following restrictions on a vector F(p) = (f_1, f_2, f_3, f_4, f_5, f_6): Trivially, f_e >= 0, f_1+f_2+...+f_6 = 6.

Furthermore,

f_2 * f_3 = 0, as 2*3 = 1*6 and 1<=n=6 and 6<=n=6, so if f_2, f_3 > 0, the value of the product isn't unique, contradiction;

f_2 < 2, 2*2 = 1*4;

f_3 * f_4 = 0 as 3*4 = 2*6. (End)

Links

Table of n, a(n) for n=1..24.

David A. Corneth, PARI program

Gerhard Kirchner, Theory and algorithm

Example

a(3) = 10 because there are 10 numbers that can be written as such a product in exactly one way:
1*1*1 = 1
1*1*2 = 2
1*1*3 = 3
1*2*2 = 4
1*2*3 = 6
2*2*2 = 8
1*3*3 = 9
2*2*3 = 12
2*3*3 = 18
3*3*3 = 27
There are 25 possible products of the numbers 1,2,3,4 (see A110713), but 9 of those products can be attained in multiple ways (e.g., 1*2*2*4 = 1*1*4*4), so a(4) = 25-9 = 16.

Mathematica

Table[Count[Split@ Sort@ Map[Times @@ # &, Union@ Map[Sort, Tuples[Range@ n, n]]], w_ /; Length@ w == 1], {n, 8}] (* Michael De Vlieger, Sep 26 2016 *)

Crossrefs

Cf. A001700, A110713.

Sequence in context: A141497 A366395 A059911 * A300017 A176760 A188396

Adjacent sequences: A160372 A160373 A160374 * A160376 A160377 A160378

Keyword

more,nonn

Author

Mats Granvik, May 11 2009

Extensions

a(7)-a(13) from Nathaniel Johnston, Nov 29 2010

a(14)-a(24) from Gerhard Kirchner, Aug 30 2016

Definition edited by N. J. A. Sloane, Sep 27 2016

Status

approved