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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. 2
1, 3, 2, 15, 10, 6, 1, 105, 70, 42, 30, 7, 5, 3, 2, 1155, 770, 462, 330, 210, 77, 55, 35, 33, 22, 21, 15, 14, 10, 6, 1, 15015, 10010, 6006, 4290, 2730, 2310, 1001, 715, 455, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The corresponding array for the numerators is given as A185972(n,m).

The sequence of row lengths of this array is 2^{n-1}=A000079(n-1), n>=1.

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

  dp(3) dp(2). Only after evaluation dp(k) becomes A007955(k).

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.

  See the W. Lang link given in A185972, and also under A007955.

LINKS

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

W. Lang: First 8 rows, also for A185972.

FORMULA

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.

EXAMPLE

[1],[3,2],[15,10,6,1],[105,70,42,30,7,5,3,2],...

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=3: A002110(3)=30 has the unique representation symbolized by  [30, 5, 3, 2]/[15, 10, 6, 1] which is

  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)/

  ((15*5*3*1)*(10*5*2*1)*(6*3*2*1)*(1)) = 30.

CROSSREFS

Cf.: A007955.

Sequence in context: A141235 A199167 A218969 * A258566 A051917 A302845

Adjacent sequences:  A185970 A185971 A185972 * A185974 A185975 A185976

KEYWORD

nonn,easy,tabf

AUTHOR

Wolfdieter Lang, Feb 08 2011

STATUS

approved

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Last modified November 22 10:59 EST 2019. Contains 329389 sequences. (Running on oeis4.)