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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). 0
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..78.

EXAMPLE

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}.

MATHEMATICA

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 *)

CROSSREFS

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

Sequence in context: A074830 A182983 A231146 * A208889 A127750 A112528

Adjacent sequences:  A287689 A287690 A287691 * A287693 A287694 A287695

KEYWORD

nonn,tabl

AUTHOR

Michael De Vlieger, Jun 15 2017

STATUS

approved

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Last modified October 28 10:30 EDT 2021. Contains 348327 sequences. (Running on oeis4.)