A353510 Square array A(n,k), n >= 1, k >= 0, with A(n,0) = n, and for k > 0, A(n,k) = A181819(A(n,k-1)), read by descending antidiagonals.

1, 1, 2, 1, 2, 3, 1, 2, 2, 4, 1, 2, 2, 3, 5, 1, 2, 2, 2, 2, 6, 1, 2, 2, 2, 2, 4, 7, 1, 2, 2, 2, 2, 3, 2, 8, 1, 2, 2, 2, 2, 2, 2, 5, 9, 1, 2, 2, 2, 2, 2, 2, 2, 3, 10, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 11, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 12, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 13, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 14

Offset: 1

Author

Antti Karttunen and Gus Wiseman, Apr 27 2022

Keywords

• nonn
• tabl

Comments

The row indexing of this array starts from 1, and the column indexing starts from 0, thus it is read by descending antidiagonals as A(1,0), A(1,1), A(2,0), A(1,2), A(2,1), A(3,0), etc.

A(n, k) gives the k-th prime shadow (the k-fold iterate of A181819) of n.

Examples

			The top left {0..6} x {1..16} corner of the array:
   1, 1, 1, 1, 1, 1, 1,
   2, 2, 2, 2, 2, 2, 2,
   3, 2, 2, 2, 2, 2, 2,
   4, 3, 2, 2, 2, 2, 2,
   5, 2, 2, 2, 2, 2, 2,
   6, 4, 3, 2, 2, 2, 2,
   7, 2, 2, 2, 2, 2, 2,
   8, 5, 2, 2, 2, 2, 2,
   9, 3, 2, 2, 2, 2, 2,
  10, 4, 3, 2, 2, 2, 2,
  11, 2, 2, 2, 2, 2, 2,
  12, 6, 4, 3, 2, 2, 2,
  13, 2, 2, 2, 2, 2, 2,
  14, 4, 3, 2, 2, 2, 2,
  15, 4, 3, 2, 2, 2, 2,
  16, 7, 2, 2, 2, 2, 2,
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11476 (rows n = 1..150, flattened)

Crossrefs

This is a full square array version of irregular triangle A325239, which after 1, lists the terms on each row only up to the first 2.

Columns 0..2: A000027, A181819, A328830.

Rows 1..2: A000012, A007395.

Programs

Mathematica

f[n_] := If[n == 1, 1, Times @@ Prime[FactorInteger[n][[All, -1]]]]; Table[Function[m, Which[m == 1, a[1, k] = 1, k == 0, a[m, 0] = m, True, Set[a[m, k], f[a[m, k - 1]]]]][n - k + 1], {n, 0, 13}, {k, n, 0, -1}] // Flatten (* Michael De Vlieger, Apr 28 2022 *)

PARI

up_to = 105;
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A353510sq(n, k) = { while(k, n = A181819(n); k--); (n); };
A353510list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, forstep(col=a-1,0,-1, i++; if(i > up_to, return(v)); v[i] = A353510sq(a-col,col))); (v); };
v353510 = A353510list(up_to);
A353510(n) = v353510[n];