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.
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
KEYWORD
nonn,tabf
AUTHOR
Michael De Vlieger, May 21 2018
STATUS
approved