A287691 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of squarefree numbers A002110(n) <= m <= (A002110(n+1)-1) such that A001221(m) = n and m is divisible by A002110(k).

1, 2, 1, 2, 4, 1, 3, 7, 8, 1, 5, 12, 23, 17, 1, 6, 16, 44, 56, 29, 1, 9, 24, 78, 130, 139, 41, 1, 9, 30, 107, 214, 351, 224, 59, 1, 11, 39, 154, 332, 707, 650, 389, 76, 1, 17, 64, 261, 598, 1475, 1637, 1489, 640, 112, 1, 21, 82, 378, 902, 2496, 3155, 3782

Offset: 0

Author

Michael De Vlieger, May 29 2017

Keywords

• nonn
• tabl

Comments

Let p_n# = A002110(n).

T(n,n) = 1 since p_n# is the only primorial divisible by p_n#.

Maxima for the first rows are {1, 2, 4, 8, 23, 56, 139, 351, 707, 1637, 3782, 8843, 18442, 38103, 77355, 177358, 387470, ...} at positions {1, 1, 2, 3, 3, 4, 5, 5, 5, 6, 7, 7, 7, 9, 10, 10, 10, ...}.

A287484(n) = sum of row n. - Michael De Vlieger, Jun 07 2017

Examples

			The triangle starts:
   n |   0    1    2    3     4     5     6     7    8    9   10
   -------------------------------------------------------------
   0 |   1
   1 |   2    1
   2 |   2    4    1
   3 |   3    7    8    1
   4 |   5   12   23   17     1
   5 |   6   16   44   56    29     1
   6 |   9   24   78  130   139    41     1
   7 |   9   30  107  214   351   224    59     1
   8 |  11   39  154  332   707   650   389    76    1
   9 |  17   64  261  598  1475  1637  1489   640  112    1
  10 |  21   82  378  902  2496  3155  3782  2505 1041  144    1
      ...
Let p_n# = A002110(n).
There are A287484(2) = 7 squarefree numbers m between p_2# = 6 and p_3# - 1 = 29: {6, 10, 14, 15, 21, 22, 26}. Of these, {15, 21} are divisible by p_0# = 1, {10, 14, 22, 26} are divisible by p_1# = 2, and {6} is divisible by p_2# = 6. Thus, T(2,k) = {2, 4, 1}.
Note that the terms {15, 21}, {10, 14, 22, 26}, and {6} pertaining to the above example appear in row n of A287483 sorted as {6, 10, 14, 15, 21, 22, 26}. - _Michael De Vlieger_, Jun 07 2017
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..230 (rows 0 <= n <= 20).

Crossrefs

Cf. A001221, A002110, A287483, A287484.

Programs

Mathematica

Table[Length /@ Split@ Sort@ Map[Block[{k = 1}, While[Divisible[#, Prime@ k], k++]; k] &, Select[Range[#, Prime[n + 1] #], And[SquareFreeQ@ #, PrimeOmega@ # == n] &] &@ Product[Prime@ i, {i, n}]], {n, 0, 6}] // Flatten (* Michael De Vlieger, May 29 2017 *)