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.

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

Author

Wolfdieter Lang, Nov 07 2021

Keywords

• nonn
• easy
• tabf

Comments

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.

Examples

			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.
		

Crossrefs

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

Programs

Mathematica

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

Formula

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.