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A185973 Array of divisor product arguments appearing in the denominator of the unique representation of primorials A002110 in terms of divisor products.

%I A185973 #9 Aug 28 2019 07:48:30
%S A185973 1,3,2,15,10,6,1,105,70,42,30,7,5,3,2,1155,770,462,330,210,77,55,35,
%T A185973 33,22,21,15,14,10,6,1,15015,10010,6006,4290,2730,2310,1001,715,455,
%U A185973 429,385,286,273,231,195,182,165,154,130,110,105,78,70,66,42,30,13,11,7,5,3,2,255255,170170,102102,72930,46410,39270,30030,17017,12155,7735,7293,6545,5005,4862,4641,3927,3315,3094,3003,2805,2618,2210,2145,2002,1870,1785,1430,1365,1326,1190,1155,1122,910,858,770,714,546,510,462,390,330,221,210,187,143,119,91,85,77,65,55,51,39,35,34,33,26,22,21,15,14,10,6,1
%N A185973 Array of divisor product arguments appearing in the denominator of the unique representation of primorials A002110 in terms of divisor products.
%C A185973 The corresponding array for the numerators is given as A185972(n,m).
%C A185973 The sequence of row lengths of this array is 2^{n-1}=A000079(n-1), n>=1.
%C A185973 The array a(n,m), m=1..2^{n-1}, n>=1, is to be read as an ordered list of numbers which give the arguments for the divisor products, called dp(). E.g., in row n=2: [3,2] stands for the ordered product
%C A185973   dp(3) dp(2). Only after evaluation dp(k) becomes A007955(k).
%C A185973 Every natural number has a unique representation in terms of divisor products dp( ) which become after evaluation A007955(k). This representation is called dpr(n). The one for the primorials n=A002110(N), N>=1, is fundamental.
%C A185973   See the W. Lang link given in A185972, and also under A007955.
%H A185973 W. Lang: <a href="/A185972/a185972.pdf">First 8 rows, also for A185972.</a>
%F A185973 a(n,m), together with A185972(n,m), is found using proposition 1 of a paper by W. Lang, given as link in A185972. In this proposition p_j has, for the application here, to be replaced by the j-th prime p(j):=A000040(j), and a() there is dp() here.
%e A185973 [1],[3,2],[15,10,6,1],[105,70,42,30,7,5,3,2],...
%e A185973 The numerator/denominator structure begins
%e A185973 [2]/[1]; [6, 1]/[3, 2]; [30, 5, 3, 2]/[15, 10, 6, 1], [210, 35, 21, 15, 14, 10, 6, 1]/[105, 70, 42, 30, 7, 5, 3, 2],...
%e A185973 n=3: A002110(3)=30 has the unique representation symbolized by  [30, 5, 3, 2]/[15, 10, 6, 1] which is
%e A185973   dp(30) dp(5) dp(3) dp(2)/dp(15) dp(10) dp(6) dp(1). Note that dp(1),although it evaluates to 1 has to be kept in the representation. This checks:  (30*15*10*6*5*3*2*1)*(5*1)*(3*1)*(2*1)/
%e A185973   ((15*5*3*1)*(10*5*2*1)*(6*3*2*1)*(1)) = 30.
%Y A185973 Cf.: A007955.
%K A185973 nonn,easy,tabf
%O A185973 1,2
%A A185973 _Wolfdieter Lang_, Feb 08 2011