A287484 -id:A287484 - cp's OEIS frontend

A287483 Irregular triangle T(n,k) read by rows: row n lists numbers m with A002110(n) <= m < A002110(n+1) such that omega(m) = n.

1, 2, 3, 5, 6, 10, 14, 15, 21, 22, 26, 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 210, 330, 390, 462, 510, 546, 570, 690, 714, 770, 798, 858, 870, 910, 930, 966, 1110, 1122, 1155, 1190, 1218, 1230, 1254, 1290

Offset: 0

Michael De Vlieger, May 25 2017

The primorial A002110(n) is the smallest squarefree number with n prime factors. Here the n-th row of the triangle is a list of squarefree numbers with n prime factors greater than and including A002110(n) but less than A002110(n+1).

A287484(n) gives row lengths.

			The sequence begins with 1 as it is equal to A002110(0) and has 0 prime factors. The first primes less than 6 come next, followed by the first squarefree semiprimes (A006881) less than 30 and the smallest terms of A033992 less than 210, etc.
Triangle begins:
n   Row n
0:   1;
1:   2,  3,  5;
2:   6, 10, 14, 15, 21,  22,  26;
3:  30, 42, 66, 70, 78, 102, 105, 110, 114, 130, ..., 195;
    ...
In each row n, the squarefree terms m must have omega(m) = n.

Michael De Vlieger, Table of n, a(n) for n = 0..10309 (rows 0 <= n <= 9).
Eric Weisstein's World of Mathematics, Primorial
Eric Weisstein's World of Mathematics, Squarefree

Cf. A001221, A002110, A005117, A006881, A033992, A287484, A287691.

Table[Select[Range[#, Prime[n + 1] # - 1] &@ Product[Prime@ i, {i, n}], And[SquareFreeQ@ #, PrimeOmega@ # == n] &], {n, 0, 4}] // Flatten

Edited by N. J. A. Sloane, Jun 05 2017

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).

Original entry on oeis.org

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

Michael De Vlieger, May 29 2017

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

			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

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

Cf. A001221, A002110, A287483, A287484.

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 *)

A287692 Triangle read by rows: T(n,k) is the greatest difference between prime factors among squarefree numbers A002110(n) <= m <= (A002110(n+1)-1) such that A001221(m) = n and m is divisible by A002110(k).

Original entry on oeis.org

3, 2, 5, 2, 3, 9, 2, 3, 5, 18, 2, 2, 4, 7, 30, 2, 2, 3, 5, 10, 42, 2, 2, 3, 4, 6, 13, 60, 2, 2, 3, 4, 5, 8, 17, 77, 2, 2, 3, 3, 4, 6, 10, 22, 113, 2, 2, 2, 3, 4, 5, 8, 12, 25, 145, 2, 2, 2, 3, 4, 5, 6, 9, 15, 32, 179, 2, 2, 2, 3, 4, 4, 6, 7, 11, 19, 36, 229

Offset: 1

Michael De Vlieger, Jun 15 2017

Let p_n# = A002110(n).
T(n,1) is the greatest index of the smallest prime divisor p of terms m in row n.
T(n,n) = A120941(n).
Consider the use of A287352 as a method for formulating squarefree numbers with n distinct prime factors. The values in row n serve as a limit beyond which we need not search further for terms p_n# <= m <= (p_(n+1)# - 1). A287352 defines a squarefree number using a sequence of nonzero positive terms, beginning with the index of the smallest prime factor, then listing differences between indexes of subsequent prime factors in order of their magnitude. We can direct increment to the largest prime index as long as the number m < p_(n+1), then increment the index before it, etc. to produce the entire tree of factors that code numbers m.

			Triangle begins:
  n\k|  1   2   3   4   5   6   7   8    9   10   11   12
---------------------------------------------------------
   1 |  3
   2 |  2   5
   3 |  2   3   9
   4 |  2   3   5  18
   5 |  2   2   4   7  30
   6 |  2   2   3   5  10  42
   7 |  2   2   3   4   6  13  60
   8 |  2   2   3   4   5   8  17  77
   9 |  2   2   3   3   4   6  10  22  113
  10 |  2   2   2   3   4   5   8  12   25  145
  11 |  2   2   2   3   4   5   6   9   15   32  179
  12 |  2   2   2   3   4   4   6   7   11   19   36  229
  ...
Let p_n# = A002110(n). For n = 2, there are A287484(2) = 7 squarefree numbers p_2# <= m <= (p_3# - 1) such that omega(m) = n. These are {6, 10, 14, 22, 26, 15, 21}. These numbers m have A287352(m) = {{1,1}, {1,2}, {1,3}, {1,4}, {1,5}, {2,1}, {2,2}} respectively; the largest values in both positions are {2,5}, thus row n = 2 of a(n) is {2,5}.

Cf. A001221, A002110, A120941, A287483, A287484, A287691.

f[n_] := If[n == 0, {{1}}, Block[{P = Product[Prime@ i, {i, n}], lim, k = 1, c, w = ConstantArray[1, n]}, lim = Prime[n + 1] P; {w}~Join~Reap[Do[w = If[k == 1, MapAt[# + 1 &, w, -k], Join[Drop[MapAt[# + 1 &, w, -k], -k + 1], ConstantArray[1, k - 1]]]; c = Times @@ Map[If[# == 0, 1, Prime@ #] &, Accumulate@ w]; If[c < lim, Sow[w]; k = 1, If[k == n, Break[], k++]], {i, Infinity}] ][[-1, 1]] ] ]; Table[Max /@ Transpose@ f@ n, {n, 14}] // Flatten (* Michael De Vlieger, Jun 15 2017 *)

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A287483 Irregular triangle T(n,k) read by rows: row n lists numbers m with A002110(n) <= m < A002110(n+1) such that omega(m) = n.

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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).

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A287692 Triangle read by rows: T(n,k) is the greatest difference between prime factors among squarefree numbers A002110(n) <= m <= (A002110(n+1)-1) such that A001221(m) = n and m is divisible by A002110(k).

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