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A122733 Least sum of n positive cubes to have exactly n prime factors, with multiplicity.

%I A122733 #22 Jan 31 2017 15:53:34
%S A122733 9,66,56,108,144,192,256,512,1024,2048,4096,8192,16384,32768,65536,
%T A122733 131072,262144,524288,1048576,2097152
%N A122733 Least sum of n positive cubes to have exactly n prime factors, with multiplicity.
%C A122733 Sequence begins with n = 2 because a(1) is undefined (sum of one positive cube cannot have exactly one prime factor, i.e., be prime).
%F A122733 a(n) = Min{x = (c_1)^3 + (c_2)^3 + ... + (c_n)^3 such that omega(x) = A001222(x) = n}.
%e A122733 a(2) = least semiprime in A003325 = 9 = 3 * 3 = 1^3 + 2^3 = A085366(1).
%e A122733 a(3) = least 3-almost prime in A003072 = 66 = 2 * 3 * 11 = 1^3 + 1^3 + 4^3 = A003072(10).
%e A122733 a(4) = least 4-almost prime in A003327 = 56 = 2^3 * 7 = 1^3 + 1^3 + 3^3 + 3^3 = A003327(10).
%e A122733 a(5) = least 5-almost prime in A003328 = 108 = 2^2 * 3^3 = 4^3 + 3^3 + 2^3 + 2^3 + 1^3 = A003328(25).
%e A122733 a(6) = least 6-almost prime in A003329 = 144 = 2^4 * 3^2 = 5^3 + 2^3 + 2^3 + 1^3 + 1^3 + 1^3 = A003329(46).
%p A122733 isSumcPosC := proc(n,c,minb)
%p A122733         local nrt ;
%p A122733         if c = 1 then nrt := iroot(n,3) ; if nrt^3 = n  and n>= minb then true; else false; end if;
%p A122733         else for b from minb do if b^3 > n then return false; end if; if isSumcPosC(n-b^3,c-1,b) then return true; end if; end do: end if;
%p A122733 end proc:
%p A122733 A122733 := proc(n)
%p A122733         for a from 1 do if numtheory[bigomega](a) = n then if isSumcPosC(a,n,1) then return a; end if; end if;
%p A122733         end do:
%p A122733 end proc:
%p A122733 for n from 2 do print(A122733(n)) ; end do: # _R. J. Mathar_, Dec 22 2010
%Y A122733 Cf. A000578, A001222, A003072, A003325, A003327, A003328, A003329, A085366.
%K A122733 nonn
%O A122733 2,1
%A A122733 _Jonathan Vos Post_, Sep 23 2006
%E A122733 a(17) from _Giovanni Resta_, Jun 13 2016
%E A122733 a(18)-a(21) more terms from _R. J. Mathar_, Jan 31 2017