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

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

Offset: 1

Author

Wolfdieter Lang, Feb 08 2011

Keywords

• nonn
• easy
• tabf

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.

Examples

			[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.
		

Links

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

Crossrefs

Cf.: A007955.

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.