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

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

Offset: 1

Author

Wolfdieter Lang, Feb 08 2011

Keywords

• nonn
• easy
• tabf

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.

Examples

			[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.
		

Links

• W. Lang: First 8 rows, also for A185973.
• Wolfdieter Lang, Divisor Product Representation for Natural Numbers.

Crossrefs

Cf. A007955.

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.