%I #18 Sep 26 2017 09:50:27
%S 1,0,27,999,573,14121,27675,68600,3528,192672,121000,158184
%N Least numbers k that can be expressed as k = a' + b', with k = a + b, in exactly n ways, where a' and b' are the arithmetic derivatives of a and b.
%C a(n) > 5*10^6 for n > 11. - _Giovanni Resta_, Sep 26 2017
%e a(0) = 1.
%e a(1) = 0 = 0' + 0'.
%e a(2) = 27 = 0' + 27' = 9' + 18'.
%e a(3) = 999 = 33' + 966' = 234' + 765' = 306' + 693'.
%e a(4) = 573 = 89' + 484' = 97' + 476' = 113' + 460' = 197' + 376'.
%e a(5) = 14121 = 5' + 14116' = 2391' + 11730' = 5310' + 8811' = 6039' + 8082' = 6066' + 8055'.
%e a(6) = 27675 = 2304' + 25371' = 7110' + 20565' = 8304' + 19371' = 8520' + 19155' = 11220' + 16455' = 12639' + 15036'.
%p with(numtheory): P:=proc(q) local a,b,c,k,n,p,x; x:=array(1..50);
%p for n from 1 to 50 do x[n]:=0; od; x[1]:=1; lprint(0,1); lprint(1,0);
%p for n from 1 to q do c:=0; for k from 0 to trunc(n/2) do
%p a:=k*add(op(2,p)/op(1,p),p=ifactors(k)[2]);
%p b:=(n-k)*add(op(2,p)/op(1,p),p=ifactors(n-k)[2]);
%p if a+b=n then c:=c+1; fi; od; if c>0 then if x[c]=0 then
%p x[c]:=n; lprint(c,x[c]); fi; fi; od; end: P(10^9);
%p # program presents a(n) not in order but as they first appear
%Y Cf. A003415.
%K nonn,hard,more
%O 0,3
%A _Paolo P. Lava_, Sep 15 2017
%E a(7), a(9)-a(11) by _Giovanni Resta_, Sep 26 2017