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A185972
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Array of divisor product arguments appearing in the numerator of the unique representation of primorials A002110 in terms of divisor products.
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2
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2, 6, 1, 30, 5, 3, 2, 210, 35, 21, 15, 14, 10, 6, 1, 2310, 385, 231, 165, 154, 110, 105, 70, 66, 42, 30, 11, 7, 5, 3, 2, 30030, 5005, 3003, 2145, 2002, 1430, 1365, 1155, 910, 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
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OFFSET
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1,1
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COMMENTS
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The corresponding array for the denominators is given as A185973(n,m).
The row lengths of this array are 2^(n-1), n>=1.
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).
Only after evaluation, dp(k) becomes A007955(k).
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.
See the W. Lang link found also under A007955.
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LINKS
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FORMULA
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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.
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EXAMPLE
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[2]; [6, 1]; [30, 5, 3, 2]; [210, 35, 21, 15, 14, 10, 6, 1];...
The numerator/denominator structure begins
[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],...
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.
Note that dp(k) should not be replaced by its value A007955(k) in the representations, only in the check.
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.
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CROSSREFS
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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STATUS
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approved
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