A348389 -id:A348389 - cp's OEIS frontend

A348392 Row sums of the irregular triangle A348389.

2, 5, 13, 18, 36, 43, 67, 85, 115, 126, 186, 199, 241, 286, 350, 367, 457, 476, 576, 639, 705, 728, 896, 946, 1024, 1105, 1245, 1274, 1484, 1515, 1675, 1774, 1876, 1981, 2269, 2306, 2420, 2537, 2817

Offset: 2

Wolfdieter Lang, Oct 31 2021

			n = 4: Compare the row of an array with all multiples of k <= n, for k = 1,2, ..., n with the row of A348389:
All multiples of k <= 4 for k = 1..4:  [1 2 3 4|2 4|3|4] with row sum A143127(4) = 23.
A348389 row 4: [2 3 4|4] with 1, 2, 3 and 4 missing: row sum is 23 - 4*5/2  = 13. Hence a(4) = A143127(4) - A000217(4).
Also: a(4) =  A348391(4) - A153485(4) = 18 - 5 = 13.

Cf. A000217, A002541, A143127, A348389.

a(n) = Sum_{m=1.. A002541(n)} A348389(n, m), for n >= 2.
a(n) = A143127(n) - A000217(n).
a(n) = A348391(n) - A153485(n).

A348390 Irregular triangle read by rows: for n >= 2 the row members a(n, m) give the proper divisors of k, followed by the multiples of k larger than k and not exceeding n, for k = 1, 2, ..., n.

Original entry on oeis.org

2, 1, 2, 3, 1, 1, 2, 3, 4, 1, 4, 1, 1, 2, 2, 3, 4, 5, 1, 4, 1, 1, 2, 1, 2, 3, 4, 5, 6, 1, 4, 6, 1, 6, 1, 2, 1, 1, 2, 3, 2, 3, 4, 5, 6, 7, 1, 4, 6, 1, 6, 1, 2, 1, 1, 2, 3, 1, 2, 3, 4, 5, 6, 7, 8, 1, 4, 6, 8, 1, 6, 1, 2, 8, 1, 1, 2, 3, 1, 1, 2, 4

Offset: 2

Wolfdieter Lang, Nov 07 2021

The length of row n is 2*A002541(n), for n >= 2.
The sum of row n is A348391(n). The sum of the proper divisors of row n is A153485(n). The sum of the multiples in row n is A348392(n). Hence, A348391(n) =  A153485(n) + A348392(n).
For k = 1 the proper divisor set is empty, and for k > floor(n/2) the set of multiples is empty.

			The irregular triangle a(n, m), m = 1, 2, ..., 2*A002541(n) begins:
(members for k = 1, 2, ..., n are separated by a vertical bar, and the proper divisors and multiples are separated by a comma)
n\m 1 2 3 4 5 6 7 8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 ...
-----------------------------------------------------------------------------------
2:  2|1
3:  2 3|1|1
4:  2 3 4|1,4|1|1 2
5:  2 3 4 5|1,4|1|1  2| 1
6:  2 3 4 5 6|1,4 6| 1, 6| 1  2| 1| 1 2 3
7:  2 3 4 5 6 7|1,4  6| 1, 6| 1  2| 1| 1  2  3| 1
8:  3 4 5 6 7 8|1,4  6  8| 1 ,6| 1  2 ,8| 1| 1  2  3| 1| 1  2  4
9:  2 3 4 5 6 7 8 9| 1, 4  6  8| 1, 6  9| 1  2, 8| 1| 1  2  3| 1| 1  2  4| 1  3
...
n = 10: 2 3 4 5 6 7 8 9 10 | 1, 4 6 8 10 | 1, 6 9 | 1 2, 8 | 1, 10 | 1 2 3 | 1 | 1 2 4 | 1 3 | 1 2 5
-----------------------------------------------------------------------------------
n = 4:  d(4, 1) = {}, m(4, 1) = {2, 3, 4}; d(4, 2) = {1}, m(4, 2) = {4}; d(4, 3) = {1}, m(4, 3) = {}; d(4, 4) = {1, 2}, m(4, 4) = {}, This explains row n = 4.

Cf. A002541, A027751, A153485, A348389, A348391, A348392.

nrows=10;Table[Flatten[Table[Join[Most[Divisors[k]],Range[2k,n,k]],{k,n}]],{n,2,nrows+1}] (* Paolo Xausa, Nov 23 2021 *)

For n >= 2 row n gives the sequence of the sequence d(n, k) of proper divisors of k (A027751(k)) followed by the sequences m(n, k) of the multiples of k, larger than k and not exceeding n (A348389), for k = 1, 2, 3, ..., n.

A348388 Irregular triangle read by rows: T(n, k) = floor((n-k)/k), for k = 1, 2, ..., floor(n/2) and n >= 2.

Original entry on oeis.org

1, 2, 3, 1, 4, 1, 5, 2, 1, 6, 2, 1, 7, 3, 1, 1, 8, 3, 2, 1, 9, 4, 2, 1, 1, 10, 4, 2, 1, 1, 11, 5, 3, 2, 1, 1, 12, 5, 3, 2, 1, 1, 13, 6, 3, 2, 1, 1, 1, 14, 6, 4, 2, 2, 1, 1, 15, 7, 4, 3, 2, 1, 1, 1, 16, 7, 4, 3, 2, 1, 1, 1, 17, 8, 5, 3, 2, 2, 1, 1, 1, 18, 8, 5, 3, 2, 2, 1, 1, 1, 19, 9, 5, 4, 3, 2, 1, 1, 1, 1

Offset: 2

Wolfdieter Lang, Oct 31 2021

This irregular triangle T(n, k) gives the number of multiples of number k, larger than k and not exceeding n, for k = 1, 2, ..., floor(n/2), for n >= 2. See A348389 for the array of these multiples.
The length of row n is floor(n/2) = A004526(n), for n >= 2.
The row sums give A002541(n). See the formula given there by Wesley Ivan Hurt, May 08 2016.
The columns give the k-fold repeated positive integers k, for k >= 1.

			The irregular triangle T(n, k) begins:
n\k   1 2 3 4 5 6 7 8 9 10 ...
------------------------------
2:    1
3:    2
4:    3 1
5:    4 1
6:    5 2 1
7:    6 2 1
8:    7 3 1 1
9:    8 3 2 1
10:   9 4 2 1 1
11:  10 4 2 1 1
12:  11 5 3 2 1 1
13:  12 5 3 2 1 1
14:  13 6 3 2 1 1 1
15:  14 6 4 2 2 1 1
16:  15 7 4 3 2 1 1 1
17:  16 7 4 3 2 1 1 1
18:  17 8 5 3 2 2 1 1 1
19:  18 8 5 3 2 2 1 1 1
20:  19 9 5 4 3 2 1 1 1  1
...

Karl-Heinz Hofmann, Table of n, a(n) for n = 2..10101

Columns k (with varying offsets): A000027, A004526, A008620, A008621, A002266, A097992, ...

Cf. A002541, A004526, A010766, A348389.

T[n_, k_] := Floor[(n - k)/k]; Table[T[n, k], {n, 2, 20}, {k, 1, Floor[n/2]}] // Flatten (* Amiram Eldar, Nov 02 2021 *)

def A348388row(n): return [(n - k) // k for k in range(1, 1 + n // 2)]
for n in range(2, 21): print(A348388row(n))  # Peter Luschny, Nov 05 2021

T(n, k) = floor((n-k)/k), for k = 1, 2, ..., floor(n/2) and n >= 2.

G.f. of column k: G(k, x) = x^(2*k)/((1 - x)*(1 - x^k)).

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A348392 Row sums of the irregular triangle A348389.

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A348390 Irregular triangle read by rows: for n >= 2 the row members a(n, m) give the proper divisors of k, followed by the multiples of k larger than k and not exceeding n, for k = 1, 2, ..., n.

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A348388 Irregular triangle read by rows: T(n, k) = floor((n-k)/k), for k = 1, 2, ..., floor(n/2) and n >= 2.

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