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A259016 A(n,k) = k-th pi-based antiderivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A259016 #24 Oct 27 2018 16:40:11
%S A259016 0,0,1,0,2,2,0,3,3,3,0,5,5,5,4,0,11,11,11,4,5,0,10,10,10,4,11,6,0,29,
%T A259016 29,29,4,10,13,7,0,78,78,78,4,29,41,6,8,0,141,141,141,4,78,35,13,19,9,
%U A259016 0,266,266,266,4,141,38,41,15,23,10,0,147,147,147,4,266,163,35,14,83,29,11
%N A259016 A(n,k) = k-th pi-based antiderivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A259016 Alois P. Heinz, <a href="/A259016/b259016.txt">Antidiagonals n = 0..20, flattened</a>
%F A259016 A(n,k) = min { m >= 0 : A258851^k(m) = n }.
%F A259016 A258850(A(n,k),k) = n.
%F A259016 A(n,k) <= A000040^k(n) for n>0.
%e A259016 A(5,3) = 29 -> 10 -> 11 -> 5.
%e A259016 A(5,4) = 78 -> 127 -> 31 -> 11 -> 5.
%e A259016 Square array A(n,k) begins:
%e A259016   0,  0,  0,   0,    0,     0,     0,      0,     0,      0, ...
%e A259016   1,  2,  3,   5,   11,    10,    29,     78,   141,    266, ...
%e A259016   2,  3,  5,  11,   10,    29,    78,    141,   266,    147, ...
%e A259016   3,  5, 11,  10,   29,    78,   141,    266,   147,    194, ...
%e A259016   4,  4,  4,   4,    4,     4,     4,      4,     4,      4, ...
%e A259016   5, 11, 10,  29,   78,   141,   266,    147,   194,   1181, ...
%e A259016   6, 13, 41,  35,   38,   163,   138,    253,   346,   1383, ...
%e A259016   7,  6, 13,  41,   35,    38,   163,    138,   253,    346, ...
%e A259016   8, 19, 15,  14,   43,   191,   201,    217,  1113,   1239, ...
%e A259016   9, 23, 83, 431, 3001, 27457, 10626, 112087, 87306, 172810, ...
%p A259016 with(numtheory):
%p A259016 d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
%p A259016 A:= proc() local t, A; t, A:= proc()-1 end, proc()-1 end;
%p A259016       proc(n, k) local h;
%p A259016         while A(n, k) = -1 do
%p A259016           t(k):= t(k)+1; h:= (d@@k)(t(k));
%p A259016           if A(h, k) = -1 then A(h, k):= t(k) fi
%p A259016         od; A(n, k)
%p A259016       end
%p A259016     end():
%p A259016 seq(seq(A(n, h-n), n=0..h), h=0..12);
%t A259016 d[n_] := If[n == 0, 0, n*Total[Last[#]*PrimePi[First[#]]/First[#]& /@ FactorInteger[n]]];
%t A259016 A[n_, k_] := For[m = 0, True, m++, If[Nest[d, m, k] == n, Return[m]]];
%t A259016 Table[A[n, k-n], {k, 0, 12}, {n, 0, k}] // Flatten (* _Jean-François Alcover_, Mar 20 2017 *)
%Y A259016 Columns k=0-3 give: A001477, A258861, A258862, A258995.
%Y A259016 Rows n=0,1,4,7,8,9 give: A000004, A258975, A010709, A259168, A259169, A259170.
%Y A259016 Cf. A000040, A000720, A258850, A258851.
%K A259016 nonn,tabl
%O A259016 0,5
%A A259016 _Alois P. Heinz_, Jun 16 2015