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

1,3

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.

Links

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

Example

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,

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 *)

Prog

(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];

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.

Sequence in context: A333486 A128628 A238966 * A334230 A275723 A198338

Adjacent sequences: A353507 A353508 A353509 * A353511 A353512 A353513

Keyword

nonn,tabl

Author

Antti Karttunen and Gus Wiseman, Apr 27 2022

Status

approved