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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).
3
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
0,2
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
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..230 (rows 0 <= n <= 20).
EXAMPLE
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
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Michael De Vlieger, May 29 2017
STATUS
approved