cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A364832 Rectangular array, read by descending antidiagonals, in which row n lists numbers m such that w(s,m) = n-1 (see comments for definition of w).

Original entry on oeis.org

1, 3, 2, 7, 6, 4, 9, 14, 12, 5, 13, 18, 28, 15, 8, 19, 26, 36, 35, 24, 10, 21, 38, 52, 45, 56, 30, 11, 27, 42, 76, 65, 72, 70, 33, 16, 31, 54, 84, 95, 104, 90, 77, 48, 17, 37, 62, 108, 105, 152, 130, 99, 112, 51, 20, 39, 74, 124, 135, 168, 190, 143, 144, 119
Offset: 1

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Author

Clark Kimberling, Aug 09 2023

Keywords

Comments

Let s be a set of primes. For each positive integer m, let w(s,m) = number of primes in s, not necessarily distinct, in the prime factorization of m. For the present sequence, we take s = A003627 = primes of the form 3k-1. Every positive integer occurs exactly once.

Examples

			Corner:
   1    3    7    9   13   19   21   27   31
   2    6   14   18   26   38   42   54   62
   4   12   28   36   52   76   84  108  124
   5   15   35   45   65   95  105  135  155
   8   24   56   72  104  152  168  216  248
  10   30   70   90  130  190  210  270  310
  11   33   77   99  143  209  231  297  341
  16   48  112  144  208  304  336  432  496
		

Crossrefs

Cf. A000040, A003627, A004612 (column 1), A343430, A364808, A346414 (row 1).

Programs

  • Mathematica
    s = Select[Prime[Range[500]], Mod[#, 3] == 2 &]; (* A003627 *)
    g[m_] := Select[FactorInteger[m], MemberQ[s, #[[1]]] &];
    h := Times @@ (#[[1]]^#[[2]]) &[Transpose[#]] & ;
    w[m_] := If[g[m] == {}, 1, h[g[m]]];
    t = Table[w[m], {m, 1, 500}] ; (* A343430 *)
    a[n_] := Flatten[Position[t, n]];
    t1 = Select[Table[a[n], {n, 1, 30}], # != {} &];
    TableForm[t1]  (* this sequence as array *)
    Table[t1[[n - k + 1, k]], {n, 12}, {k, n, 1, -1}] // Flatten  (* this sequence *)

Formula

(row n) = A004612(n)*(row 1).
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