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A258850 A(n,k) = k-th pi-based arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A258850 #26 Oct 24 2018 16:23:18
%S A258850 0,0,1,0,0,2,0,0,1,3,0,0,0,2,4,0,0,0,1,4,5,0,0,0,0,4,3,6,0,0,0,0,4,2,
%T A258850 7,7,0,0,0,0,4,1,4,4,8,0,0,0,0,4,0,4,4,12,9,0,0,0,0,4,0,4,4,20,12,10,
%U A258850 0,0,0,0,4,0,4,4,32,20,11,11,0,0,0,0,4,0,4,4,80,32,5,5,12
%N A258850 A(n,k) = k-th pi-based arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A258850 Alois P. Heinz, <a href="/A258850/b258850.txt">Antidiagonals n = 0..31, flattened</a>
%F A258850 A(n,k) = A258851^k(n).
%F A258850 A(A259016(n,k),k) = n.
%F A258850 A(A258975(n),n) = 1.
%e A258850 Square array A(n,k) begins:
%e A258850   0,  0,  0,  0,  0,   0,   0,    0,     0,     0, ...
%e A258850   1,  0,  0,  0,  0,   0,   0,    0,     0,     0, ...
%e A258850   2,  1,  0,  0,  0,   0,   0,    0,     0,     0, ...
%e A258850   3,  2,  1,  0,  0,   0,   0,    0,     0,     0, ...
%e A258850   4,  4,  4,  4,  4,   4,   4,    4,     4,     4, ...
%e A258850   5,  3,  2,  1,  0,   0,   0,    0,     0,     0, ...
%e A258850   6,  7,  4,  4,  4,   4,   4,    4,     4,     4, ...
%e A258850   7,  4,  4,  4,  4,   4,   4,    4,     4,     4, ...
%e A258850   8, 12, 20, 32, 80, 208, 512, 2304, 12288, 81920, ...
%e A258850   9, 12, 20, 32, 80, 208, 512, 2304, 12288, 81920, ...
%p A258850 with(numtheory):
%p A258850 d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
%p A258850 A:= proc(n, k) option remember; `if`(k=0, n, d(A(n, k-1))) end:
%p A258850 seq(seq(A(n, h-n), n=0..h), h=0..14);
%t A258850 d[n_] := n*Total[Last[#]*PrimePi[First[#]]/First[#]& /@ FactorInteger[n]]; d[0] = 0;
%t A258850 A[n_, k_] := A[n, k] = If[k == 0, n, d[A[n, k-1]]];
%t A258850 Table[Table[A[n, h-n], {n, 0, h}], {h, 0, 14}] // Flatten (* _Jean-François Alcover_, Apr 24 2016, adapted from Maple *)
%Y A258850 Columns k=0-10 give: A001477, A258851, A258852, A258853, A258854, A258855, A258856, A258857, A258858, A258859, A258860.
%Y A258850 Rows n=0,1,4,8 give: A000004, A000007, A010709, A258848.
%Y A258850 Antidiagonal sums give A258847.
%Y A258850 Main diagonal gives A258849.
%Y A258850 Cf. A000720, A258975, A259016.
%K A258850 nonn,tabl
%O A258850 0,6
%A A258850 _Alois P. Heinz_, Jun 12 2015