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A283368 Irregular triangle read by rows: T(n,k) = number of heights for the horizontal elements of the Dyck paths for the symmetric representation of sigma(n) that are listed in the corresponding positions of the triangle of A259176. 2
1, 2, 3, 2, 4, 3, 5, 3, 6, 5, 4, 7, 5, 4, 8, 6, 5, 9, 7, 5, 10, 8, 7, 6, 11, 8, 7, 6, 12, 10, 9, 7, 13, 10, 9, 7, 14, 11, 9, 8, 15, 12, 11, 10, 8, 16, 13, 12, 11, 9, 17, 13, 12, 11, 9, 18, 15, 13, 12, 10, 19, 15, 13, 12, 10, 20, 16, 15, 13, 11, 21, 17, 16, 15, 14, 11 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The dot product of the n-th row of this triangle and the n-th row of triangle A259176 equals A024916(n), the sum of all divisors of numbers 1 through n (true for all n <= 20000); the value is the sum of the rectangles between the x-axis and the horizontal legs of the symmetric representation of sigma(n). This is the companion computation to A283367.

LINKS

Table of n, a(n) for n=1..76.

FORMULA

T(n,k) = n - sum_{i=1..k-1} f(n, 2*i) where f is defined in A237593.

A024916(n) = sum_{i=1..row(n)} (T(n,i))*S(n,i)) where S(n,i) refers to the triangle of A259176 and row(n) = floor(sqrt(8*n+1)-1)/2).

EXAMPLE

The first horizontal leg of the symmetric representation of sigma(15) is at y-coordinate 15 and has length 8, and row 15 has 5 entries so that T(15,1) = 15 and T(15,5) = 8.

The first 16 rows of the irregular triangle:

1

2

3    2

4    3

5    3

6    5    4

7    5    4

8    6    5

9    7    5

10    8    7    6

11    8    7    6

12   10    9    7

13   10    9    7

14   11    9    8

15   12   11   10    8

16   13   12   11    9

MATHEMATICA

(* function f[n, k] and its support functions are defined in A237593 *)

a283368[n_, k_] := n - Sum[f[n, 2i], {i, k-1}]

TableForm[Table[a283368[n, k], {n, 1, 16}, {k, 1, row[n]}]] (* triangle *)

Flatten[Table[a283368[n, k], {n, 1, 21}, {k, 1, row[n]}]] (* sequence data *)

CROSSREFS

Cf. A024916, A237593, A259176, A259177, A283367.

Sequence in context: A157893 A331253 A264116 * A199474 A246694 A054384

Adjacent sequences:  A283365 A283366 A283367 * A283369 A283370 A283371

KEYWORD

nonn,tabf

AUTHOR

Hartmut F. W. Hoft, Mar 06 2017

STATUS

approved

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Last modified July 28 19:33 EDT 2021. Contains 346335 sequences. (Running on oeis4.)