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A185972 Array of divisor product arguments appearing in the numerator of the unique representation of primorials A002110 in terms of divisor products. 2


%S 2,6,1,30,5,3,2,210,35,21,15,14,10,6,1,2310,385,231,165,154,110,105,

%T 70,66,42,30,11,7,5,3,2,30030,5005,3003,2145,2002,1430,1365,1155,910,

%U 858,770,546,462,390,330,210,143,91,77,65,55,39,35,33,26,22,21,15,14,10,6,1,510510,85085,51051,36465,34034,24310,23205,19635,15470,15015,14586,13090,10010,9282,7854,6630,6006,5610,4290,3570,2730,2431,2310,1547,1309,1105,1001,935,715,663,595,561,455,442,429,385,374,357,286,273,255,238,231,195,182,170,165,154,130,110,105,102,78,70,66,42,30,17,13,11,7,5,3,2

%N Array of divisor product arguments appearing in the numerator of the unique representation of primorials A002110 in terms of divisor products.

%C The corresponding array for the denominators is given as A185973(n,m).

%C The row lengths of this array are 2^(n-1), n>=1.

%C The array a(n,m), m=1..2^{n-1}, n>=1, is an ordered list of numbers which give the arguments for the divisor products, called dp(). E.g., in the row n=2, [6,1] represents the ordered product dp(6)*dp(1).

%C Only after evaluation, dp(k) becomes A007955(k).

%C Every natural number has a unique representation in terms of products of divisors 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 See the W. Lang link found also under A007955.

%H W. Lang: <a href="/A185972/a185972.pdf">First 8 rows, also for A185973.</a>

%H Wolfdieter Lang, <a href="/A007955/a007955.pdf">Divisor Product Representation for Natural Numbers.</a>

%F a(n,m), together with A185973(n,m), is found using proposition 1 of a paper by W. Lang, given as link above. In this proposition p_j has, for this application, to be replaced by the j-th prime p(j)=A000040(j), and a() there is dp() here.

%e [2]; [6, 1]; [30, 5, 3, 2]; [210, 35, 21, 15, 14, 10, 6, 1];...

%e The numerator/denominator structure begins

%e [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 n=1: A002110(1)=2 has the unique representation dp(2)/dp(1), with dp(k) the product of divisors of k. This checks when evaluated: (2*1)/(1) = 2.

%e Note that dp(k) should not be replaced by its value A007955(k) in the representations, only in the check.

%e n=2: A002110(2)=6 has the unique representation dp(6)*dp(2)/(dp(3)*dp(2)) which checks: (6*3*2*1)*(2*1)/((3*1)*(2*1)) = 6.

%Y Cf. A007955.

%K nonn,easy,tabf

%O 1,1

%A _Wolfdieter Lang_, Feb 08 2011

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