A240698 Partial sums of divisors of n, cf. A027750.

1, 1, 3, 1, 4, 1, 3, 7, 1, 6, 1, 3, 6, 12, 1, 8, 1, 3, 7, 15, 1, 4, 13, 1, 3, 8, 18, 1, 12, 1, 3, 6, 10, 16, 28, 1, 14, 1, 3, 10, 24, 1, 4, 9, 24, 1, 3, 7, 15, 31, 1, 18, 1, 3, 6, 12, 21, 39, 1, 20, 1, 3, 7, 12, 22, 42, 1, 4, 11, 32, 1, 3, 14, 36, 1, 24, 1

Offset: 1

Author

Reinhard Zumkeller, Apr 10 2014

Keywords

• nonn
• tabf

Comments

Triangle read by rows in which row n lists the partial sums of divisors of n. - Omar E. Pol, Apr 12 2014

Examples

			.    n |  n-th row of A240698   |  n-th row of A027750
.  ----+------------------------+---------------------
.    1 |  1                     |  1
.    2 |  1, 3                  |  1, 2
.    3 |  1, 4                  |  1, 3
.    4 |  1, 3, 7               |  1, 2, 4
.    5 |  1, 6                  |  1, 5
.    6 |  1, 3, 6, 12           |  1, 2, 3, 6
.    7 |  1, 8                  |  1, 7
.    8 |  1, 3, 7, 15           |  1, 2, 4, 8
.    9 |  1, 4, 13              |  1, 3, 9
.   10 |  1, 3, 8, 18           |  1, 2, 5, 10
.   11 |  1, 12                 |  1, 11
.   12 |  1, 3, 6, 10, 16, 28   |  1, 2, 3, 4, 6, 12
.   13 |  1, 14                 |  1, 13 .
		

Links

• Reinhard Zumkeller, Rows n = 1..1000 of table, flattened

Crossrefs

Cf. A000005 (row lengths), A240694.

Cf. A000203, A001065, A027750, A064510, A194472.

Programs

Haskell

a240698 n k = a240698_tabf !! (n-1) !! (k-1)
a240698_row n = a240698_tabf !! (n-1)
a240698_tabf = map (scanl1 (+)) a027750_tabf

Mathematica

Table[Accumulate[Divisors[n]],{n,30}]//Flatten (* Harvey P. Dale, Dec 30 2019 *)

PARI

row(n) = my(d=divisors(n)); vector(#d, k, sum(i=1, k, d[i])); \\ Michel Marcus, Jan 24 2022

Formula

T(n,1) = 1, T(n,k) = T(n,k-1) + A027750(n,k), 1 < k <= n.

T(n,1) = 1;

T(n,A000005(n)) = A000203(n);

T(n,A000005(n)-1) = A001065(n), n > 1.