A160375 - OEIS

%I #36 Jan 10 2017 07:12:12

%S 1,3,10,16,61,81,337,477,601,901,4291,5798,27314,33671,45732,59397,

%T 299745,421363,2090647,2739022,4597263,5401826,27510715,23666955

%N 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.

%C Number of combinations as in A001700.

%C From _David A. Corneth_, Sep 26 2016: (Start)

%C 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.

%C 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.

%C 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).

%C 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.

%C Furthermore,

%C 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;

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

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

%H David A. Corneth, <a href="/A160375/a160375.gp.txt">PARI program</a>

%H Gerhard Kirchner, <a href="/A160375/a160375.txt">Theory and algorithm</a>

%e a(3) = 10 because there are 10 numbers that can be written as such a product in exactly one way:

%e 1*1*1 = 1

%e 1*1*2 = 2

%e 1*1*3 = 3

%e 1*2*2 = 4

%e 1*2*3 = 6

%e 2*2*2 = 8

%e 1*3*3 = 9

%e 2*2*3 = 12

%e 2*3*3 = 18

%e 3*3*3 = 27

%e 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.

%t 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 *)

%Y Cf. A001700, A110713.

%K more,nonn

%O 1,2

%A _Mats Granvik_, May 11 2009

%E a(7)-a(13) from _Nathaniel Johnston_, Nov 29 2010

%E a(14)-a(24) from _Gerhard Kirchner_, Aug 30 2016

%E Definition edited by _N. J. A. Sloane_, Sep 27 2016