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%I #11 Aug 03 2014 14:01:28
%S 1,2,4,5,7,8,15,23,31,87,111,119
%N Numbers that are not the sum of three powerful numbers (A001694).
%C Heath-Brown shows that this sequence is finite, resolving a conjecture of Erdos. Presumably a(12) = 119 is the last term.
%D D. R. Heath-Brown, "Sums of three square-full numbers". Number theory, Vol. I (Budapest, 1987), pp. 163-171, Colloq. Math. Soc. János Bolyai, 51, North-Holland, Amsterdam, 1990.
%D D. R. Heath-Brown, "Ternary quadratic forms and sums of three square-full numbers". Séminaire de Théorie des Nombres, Paris 1986-87, pp. 137-163, Progr. Math., 75, Birkhäuser Boston, Boston, MA, 1988.
%H P. Erdos, <a href="http://www.renyi.hu/~p_erdos/1976-39.pdf">Problems and results on number theoretic properties of consecutive integers and related questions</a>, Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975), Congress. Numer. XVI (1976), pp. 25-44.
%t powerfulQ[n_] := n == 1 || Min[Last /@ FactorInteger[n]] > 1; nn = 1000; pow = Select[Range[nn], powerfulQ]; Complement[Range[nn], Select[Union[Flatten[Outer[Plus, pow, pow, pow]]], # <= nn &]] (* _T. D. Noe_, Mar 02 2011 *)
%Y Proper subsequence of A135367.
%Y Cf. A076871, A135693.
%K nonn,fini,full
%O 1,2
%A _Charles R Greathouse IV_, Mar 02 2011