A304886 Irregular triangle where row n contains indices k where the product of A002110(k) = A025487(n).

0, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 1, 1, 1, 1, 1, 2, 4, 2, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2

Offset: 1

Author

Michael De Vlieger, May 21 2018

Keywords

• nonn
• tabf

Comments

Row n consists of terms k such that A025487(n) = the product of primorials p_k#, the k in row n written least to greatest k.

For m = A025487(n) in A000079 (i.e., m is an integer power of 2), row n contains A000079(m) 1s.

For m = A025487(n) in A002110 (i.e., m is a primorial) row n contains a single term k that is the index of m in A002110.

Examples

			Triangle begins as in rightmost column, which lists the terms that occur on row n. Maximum value of each row is given by A061394(n).
   n  A025487(n)   Row n
--------------------------------
   1        1      0
   2        2      1
   3        4      1,1
   4        6      2
   5        8      1,1,1
   6       12      1,2
   7       16      1,1,1,1
   8       24      1,1,2
   9       30      3
  10       32      1,1,1,1,1
  11       36      2,2
  12       48      1,1,1,2
  13       60      1,3
  14       64      1,1,1,1,1,1
  15       72      1,2,2
  16       96      1,1,1,1,2
  17      120      1,1,3
  18      128      1,1,1,1,1,1,1
  19      144      1,1,2,2
  20      180      2,3
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..8600
• Michael De Vlieger, Concordance of A025487, A051282, A061394, and A304886
• Michael De Vlieger, Indices of primorials whose product is highly composite
• Michael De Vlieger, Indices of primorials whose product is superabundant

Crossrefs

Cf. A025487, A051282 (row lengths), A061394 (row maximum), A124832, A181815.

Cf. also A307056.

Programs

Mathematica

(* Simple (A025487(n) < 10^5): *)
{{0}}~Join~Map[With[{w = #}, Reverse@ Array[Function[k, Count[w, _?(# >= k &)] ], Max@ w]] &, Select[Array[{#, FactorInteger[#][[All, -1]]} &, 400], Times @@ Boole@ {#1 == Times @@ MapIndexed[Prime[First@ #2]^#1 &, #3], #2 == #3} == 1 & @@ {#1, #2, Sort[#2, Greater]} & @@ # &][[All, -1]] ]
(* Efficient (A025487(n) < 10^23): *)
f[n_] := Block[{ww, g, h},
  g[x_] := Apply[Times,
    MapIndexed[Prime[First@ #2]^#1 &, x]];
  h[x_] := Reverse@
    Array[Function[k, Count[x, _?(# >= k &)] ], Max@ x];
  ww = NestList[Append[#, 1] &, {1}, # - 1] &[-2 +
     Length@ NestWhileList[NextPrime@ # &, 1,
     Times @@ {##} <= n &, All] ];
  Map[h, SortBy[Flatten[#, 1], g]] &@
   Map[Block[{w = #, k = 1},
      Apply[
         Join, {{ConstantArray[1, Length@ w]},
           If[Length@ # == 0, #, #[[1]]] }] &@ Reap[
         Do[
          If[# < n,
            Sow[w]; k = 1,
             If[k >= Length@ w, Break[], k++]] &@
               g@ Set[w,
               If[k == 1,
                 MapAt[# + 1 &, w, k],
                 PadLeft[#, Length@ w, First@ #] &@
                   Drop[MapAt[# + Boole[i > 1] &, w, k],
                    k - 1] ]], {i, Infinity}] ][[-1]] ] &, ww]]; {{0}}~Join~f@ 400

Formula

For row n > 1, Product_{k=1..A051282(n)} A000040(T(n,k)) = A181815(n). [Product of primes indexed by nonzero terms of row n is equal to A181815(n)] - Antti Karttunen, Dec 28 2019