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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.

%I A160375 #36 Jan 10 2017 07:12:12
%S A160375 1,3,10,16,61,81,337,477,601,901,4291,5798,27314,33671,45732,59397,
%T A160375 299745,421363,2090647,2739022,4597263,5401826,27510715,23666955
%N 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.
%C A160375 Number of combinations as in A001700.
%C A160375 From _David A. Corneth_, Sep 26 2016: (Start)
%C A160375 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 A160375 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 A160375 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 A160375 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 A160375 Furthermore,
%C A160375 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 A160375 f_2 < 2, 2*2 = 1*4;
%C A160375 f_3 * f_4 = 0 as 3*4 = 2*6. (End)
%H A160375 David A. Corneth, <a href="/A160375/a160375.gp.txt">PARI program</a>
%H A160375 Gerhard Kirchner, <a href="/A160375/a160375.txt">Theory and algorithm</a>
%e A160375 a(3) = 10 because there are 10 numbers that can be written as such a product in exactly one way:
%e A160375 1*1*1 = 1
%e A160375 1*1*2 = 2
%e A160375 1*1*3 = 3
%e A160375 1*2*2 = 4
%e A160375 1*2*3 = 6
%e A160375 2*2*2 = 8
%e A160375 1*3*3 = 9
%e A160375 2*2*3 = 12
%e A160375 2*3*3 = 18
%e A160375 3*3*3 = 27
%e A160375 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 A160375 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 A160375 Cf. A001700, A110713.
%K A160375 more,nonn
%O A160375 1,2
%A A160375 _Mats Granvik_, May 11 2009
%E A160375 a(7)-a(13) from _Nathaniel Johnston_, Nov 29 2010
%E A160375 a(14)-a(24) from _Gerhard Kirchner_, Aug 30 2016
%E A160375 Definition edited by _N. J. A. Sloane_, Sep 27 2016