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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.
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%I #12 Apr 28 2022 09:43:14

%S 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,

%T 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,

%U 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

%N 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.

%C 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.

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

%H Michael De Vlieger, <a href="/A353510/b353510.txt">Table of n, a(n) for n = 1..11476</a> (rows n = 1..150, flattened)

%e The top left {0..6} x {1..16} corner of the array:

%e 1, 1, 1, 1, 1, 1, 1,

%e 2, 2, 2, 2, 2, 2, 2,

%e 3, 2, 2, 2, 2, 2, 2,

%e 4, 3, 2, 2, 2, 2, 2,

%e 5, 2, 2, 2, 2, 2, 2,

%e 6, 4, 3, 2, 2, 2, 2,

%e 7, 2, 2, 2, 2, 2, 2,

%e 8, 5, 2, 2, 2, 2, 2,

%e 9, 3, 2, 2, 2, 2, 2,

%e 10, 4, 3, 2, 2, 2, 2,

%e 11, 2, 2, 2, 2, 2, 2,

%e 12, 6, 4, 3, 2, 2, 2,

%e 13, 2, 2, 2, 2, 2, 2,

%e 14, 4, 3, 2, 2, 2, 2,

%e 15, 4, 3, 2, 2, 2, 2,

%e 16, 7, 2, 2, 2, 2, 2,

%t 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 *)

%o (PARI)

%o up_to = 105;

%o A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));

%o A353510sq(n, k) = { while(k, n = A181819(n); k--); (n); };

%o 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); };

%o v353510 = A353510list(up_to);

%o A353510(n) = v353510[n];

%Y 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.

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

%Y Rows 1..2: A000012, A007395.

%K nonn,tabl

%O 1,3

%A _Antti Karttunen_ and _Gus Wiseman_, Apr 27 2022