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%I #15 Dec 22 2021 15:54:54
%S 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,
%T 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,
%U 8,1,6,1,2,8,1,1,2,3,1,1,2,4
%N 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.
%C The length of row n is 2*A002541(n), for n >= 2.
%C 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).
%C For k = 1 the proper divisor set is empty, and for k > floor(n/2) the set of multiples is empty.
%F 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.
%e The irregular triangle a(n, m), m = 1, 2, ..., 2*A002541(n) begins:
%e (members for k = 1, 2, ..., n are separated by a vertical bar, and the proper divisors and multiples are separated by a comma)
%e 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 ...
%e -----------------------------------------------------------------------------------
%e 2: 2|1
%e 3: 2 3|1|1
%e 4: 2 3 4|1,4|1|1 2
%e 5: 2 3 4 5|1,4|1|1 2| 1
%e 6: 2 3 4 5 6|1,4 6| 1, 6| 1 2| 1| 1 2 3
%e 7: 2 3 4 5 6 7|1,4 6| 1, 6| 1 2| 1| 1 2 3| 1
%e 8: 3 4 5 6 7 8|1,4 6 8| 1 ,6| 1 2 ,8| 1| 1 2 3| 1| 1 2 4
%e 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
%e ...
%e 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
%e -----------------------------------------------------------------------------------
%e 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.
%t nrows=10;Table[Flatten[Table[Join[Most[Divisors[k]],Range[2k,n,k]],{k,n}]],{n,2,nrows+1}] (* _Paolo Xausa_, Nov 23 2021 *)
%Y Cf. A002541, A027751, A153485, A348389, A348391, A348392.
%K nonn,easy,tabf
%O 2,1
%A _Wolfdieter Lang_, Nov 07 2021
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