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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
1, 3, 10, 16, 61, 81, 337, 477, 601, 901, 4291, 5798, 27314, 33671, 45732, 59397, 299745, 421363, 2090647, 2739022, 4597263, 5401826, 27510715, 23666955 (list; graph; refs; listen; history; text; internal format)
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: A083684 A141497 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

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Last modified March 26 01:08 EDT 2019. Contains 321479 sequences. (Running on oeis4.)