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

1,5

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.

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

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

Example

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
  ...

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

Crossrefs

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

Cf. also A307056.

Sequence in context: A296081 A074064 A275215 * A352080 A295632 A139549

Adjacent sequences: A304883 A304884 A304885 * A304887 A304888 A304889

Keyword

nonn,tabf

Author

Michael De Vlieger, May 21 2018

Status

approved