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A317389 a(0) = 0, a(1) = a(2) = 1; for n >= 3, a(n) = freq(a(g(n)),n)*freq(a(n-g(n)),n) where g = A000195 and freq(i, j) is the number of times i appears in the terms a(0) .. a(j-1).

%I A317389 #12 Nov 27 2020 06:31:44
%S A317389 0,1,1,4,2,2,4,4,6,6,4,4,10,10,4,4,14,14,4,4,18,81,81,9,18,18,9,27,27,
%T A317389 18,18,18,54,54,54,27,27,27,45,45,45,27,27,27,72,72,72,27,27,27,99,99,
%U A317389 99,27,27,6,6,26,26,8,8,4,4,4,4,26,26,26,26,12,12,12,12,8,8,8,8,12,12,12,12,16,16,16,16,8,8,8,8,20,20,20
%N A317389 a(0) = 0, a(1) = a(2) = 1; for n >= 3, a(n) = freq(a(g(n)),n)*freq(a(n-g(n)),n) where g = A000195 and freq(i, j) is the number of times i appears in the terms a(0) .. a(j-1).
%H A317389 Alois P. Heinz, <a href="/A317389/b317389.txt">Table of n, a(n) for n = 0..65536</a>
%p A317389 b:= proc() 0 end:
%p A317389 a:= proc(n) option remember; local t;
%p A317389       t:= `if`(n<3, n*(3-n)/2, b(a(ilog(n)))*b(a(n-ilog(n))));
%p A317389       b(t):= b(t)+1; t
%p A317389     end:
%p A317389 seq(a(n), n=0..200);  # _Alois P. Heinz_, Jul 27 2018
%t A317389 b[_] = 0;
%t A317389 a[n_] := a[n] = Module[{t}, t = If[n<3, n(3-n)/2, b[a[Floor@Log[n]]] b[a[n - Floor@Log[n]]]]; b[t] = b[t]+1; t];
%t A317389 a /@ Range[0, 200] (* _Jean-François Alcover_, Nov 27 2020, after _Alois P. Heinz_ *)
%Y A317389 Cf. A317359.
%K A317389 nonn,look
%O A317389 0,4
%A A317389 _Altug Alkan_, Jul 27 2018