A193829 Irregular triangle read by rows in which row n lists the differences between consecutive divisors of n.

1, 2, 1, 2, 4, 1, 1, 3, 6, 1, 2, 4, 2, 6, 1, 3, 5, 10, 1, 1, 1, 2, 6, 12, 1, 5, 7, 2, 2, 10, 1, 2, 4, 8, 16, 1, 1, 3, 3, 9, 18, 1, 2, 1, 5, 10, 2, 4, 14, 1, 9, 11, 22, 1, 1, 1, 2, 2, 4, 12, 4, 20, 1, 11, 13, 2, 6, 18, 1, 2, 3, 7, 14, 28, 1, 1, 2, 1, 4, 5, 15, 30

Offset: 2

Author

Omar E. Pol, Aug 31 2011

Keywords

• nonn
• tabf
• easy

Comments

The sum of row n gives A000027(n-1). The product of row n gives A057449(n). Row n has length A032741(n). The final term of row n is A060681(n). It appears that the first term of row n is A057237(n).

Examples

			Written as a triangle:
1,
2,
1, 2,
4,
1, 1, 3,
6,
1, 2, 4,
2, 6,
1, 3, 5,
10,
1, 1, 1, 2, 6
		

Links

• Alois P. Heinz, Rows n = 2..1600, flattened
• Omar E. Pol, Illustration of divisors of n
• Wikipedia, Table of divisors

Crossrefs

Cf. A027750, A032742.

Cf. A060682 (distinct terms per row), A060680 (row minima), A060681 (row maxima).

Programs

Haskell

import Data.List (genericIndex)
a193829 n k = genericIndex a193829_tabf (n - 1) !! (k - 1)
a193829_row n = genericIndex a193829_tabf (n - 1)
a193829_tabf = zipWith (zipWith (-))
                       (map tail a027750_tabf') a027750_tabf'
-- Reinhard Zumkeller, Jun 25 2015, Jun 23 2013

Mathematica

Flatten[Table[Differences[Divisors[n]], {n, 2, 30}]] (* T. D. Noe, Aug 31 2011 *)

Formula

T(n,k) = A027750(n,k+1)-A027750(n,k). - R. J. Mathar, Sep 01 2011