A384000 -id:A384000 - cp's OEIS frontend

A384001 Irregular triangle T(n,j,k), j = 1..A024718(n), k = 1..n, where row 1 = {(0), (1)}, and row n = union of n-tuples whose sum s < n, and the n-tuples formed by appending s to the (n-1)-tuples in row n-1.

0, 1, 0, 0, 0, 1, 0, 2, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 1, 0, 0, 1, 1, 0, 1, 2, 0, 2, 0, 0, 2, 1, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 4, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 2

Offset: 1

Michael De Vlieger, May 21 2025

Terms in row n are sorted lexicographically.
Row n is created by finding n-tuples w with elements from 0..n-1, taking only those w whose sums are less than n.
For example, row n = 3 contains 3-tuples w that have elements from 0..2, i.e., {(0,0,0), (0,0,1), (0,0,2), (0,1,0), (0,1,1), (0,2,0), (1,0,0), (1,0,1), (1,1,0), (2,0,0)}.
Let s be the sum of w. Then we take all elements w of row n-1 and append n-s to w to obtain certain 3-tuples with elements from 0..n whose sum s = n.
Continuing the example, row 2 = {(0,0), (0,1), (0,2), (1,0), (1,1)}, which, adding n-s to the right end gives {(0,0,3), (0,1,2), (0,2,1), (1,0,2), (1,1,1)}.
Let p_i be the i-th smallest prime divisor of N = A384000(n) (where i is not necessarily the i-th prime). Then, the terms m in row N of A162306 are of the form m = Product_{i..n} p_i^T(n,j,n-k+1). For instance, for N = 6, we have row 6 of A162306 = {1, 2, 3, 4, 6}, which is {2^0*3^0, 2^1*3^0, 2^2*3^0, 2^0*3^1, 2^1*3^1} = {1, 2, 4, 3, 6}, sorted.

			Table begins:
  1:   (0), (1);
  2:   (0, 0), (0, 1), (0, 2), (1, 0), (1, 1);
  3:   (0,0,0), (0,0,1), (0,0,2), (0,0,3), (0,1,0),
       (0,1,1), (0,1,2), (0,2,0), (0,2,1), (1,0,0),
       (1,0,1), (1,0,2), (1,1,0), (1,1,1), (2,0,0)
  etc.
Row 2 arranged as a rank 2 table, concatenating T(2,j,k), k = 1..2:
00   10   20
01   11
.
Row 3 arranged as a rank 3 table, concatenating T(3,j,k), k = 1..3:
000  001  002  003     100  101  102    200
010  011  012          110  111
020  021

Cf. A001700, A024718, A384000, A384002.

nn = 4; w[0] = {{0}};
Do[If[n == 1, Set[w[1], {{0}, {1}}],
  Set[w[n], Union@ Join[Select[Tuples[Range[0, n - 1], n], Total[#] < n &],
    Map[Append[#, n - Total[#]] &, w[n - 1] ] ] ] ], {n, nn}];
Flatten@ Array[w, nn]

Length of row n = n*A024718(n).

A384960 a(n) = smallest sphenic number k such that A010846(k) = n.

Original entry on oeis.org

1001, 105, 231, 30, 42, 70, 110, 66, 78, 170, 102, 114, 138, 370, 174, 826, 222, 246, 258, 318, 354, 402, 438, 498, 534, 582, 654, 762, 786, 894, 978, 1038, 1158, 1338, 1506, 1542, 1758, 1986, 2082, 2202, 2334, 2598, 2922, 3126, 3462, 3918, 4098, 4398, 4614, 5262

Offset: 15

Michael De Vlieger, Jul 06 2025

nonn

a(1) = A384000(3) = 1001; A010846(1001) = A024718(3) = 15; 1001 is the smallest number k with 3 distinct prime factors that has the smallest possible number of terms in row k of A162306, i.e., m <= k such that rad(m) | k.

For n > 30, 6 | a(n).

			Table of a(n) indicating prime factors and S, where S = {ceiling(log_p a(n))} for all primes p that divide a(n), in order of the magnitude of p.
                                Prime power factor
                                    1111223344455
 n  m=a(n) pi(facs(m))    S     23571379391713739
-------------------------------------------------
15   1001   4.5.6       4.3.3   ...111
16    105   2.3.4       5.3.3   .111
17    231   2.4.5       5.3.3   .1.11
18     30   1.2.3       5.4.3   111
19     42   1.2.4       6.4.2   11.1
20     70   1.3.4       7.3.3   1.11
21    110   1.3.5       7.3.2   1.1.1
22     66   1.2.5       7.4.2   11..1
23     78   1.2.6       7.4.2   11...1
24    170   1.3.7       8.4.2   1.1...1
25    102   1.2.7       7.5.2   11....1
26    114   1.2.8       7.5.2   11.....1
27    138   1.2.9       8.5.2   11......1
28    370   1.3.12      9.4.2   1.1........1
29    174   1.2.10      8.5.2   11.......1
30    826   1.4.17     10.4.2   1..1............1
31    222   1.2.12      8.5.2   11.........1
32    246   1.2.13      8.6.2   11..........1
33    258   1.2.14      9.6.2   11...........1
34    318   1.2.16      9.6.2   11.............1

Michael De Vlieger, Table of n, a(n) for n = 15..300
Michael De Vlieger, Plot of terms k = p^a*q^b*r^c, primes p < q < r, in row a(n) of A162306, n = 15..50, at (x,y,z) = (a,b,c). For a(n) there are n blocks in each diagram.
Michael De Vlieger, Mathematica code.

Cf. A007304, A007947, A010846, A024718, A162306, A384000.

(* See Mathematica code link for function definitions for kappaomega and theta *)
s =  kappaomega[3, 6000]; t = Map[theta, s];
Map[s[[FirstPosition[t, #][[1]] ]] &, Union[t]]

cp's OEIS Frontend

A384001 Irregular triangle T(n,j,k), j = 1..A024718(n), k = 1..n, where row 1 = {(0), (1)}, and row n = union of n-tuples whose sum s < n, and the n-tuples formed by appending s to the (n-1)-tuples in row n-1.

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