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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
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..127.

W. Lang: First 8 rows, also for A185973.

W. Lang: Divisor Product Representation for Natural Numbers.

FORMULA

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.

EXAMPLE

[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.

CROSSREFS

Cf. A007955.

Sequence in context: A027642 A249306 A117214 * A182918 A134301 A168294

Adjacent sequences:  A185969 A185970 A185971 * A185973 A185974 A185975

KEYWORD

nonn,easy,tabf

AUTHOR

Wolfdieter Lang, Feb 08 2011

STATUS

approved

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Last modified October 16 14:35 EDT 2018. Contains 316263 sequences. (Running on oeis4.)