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A210470 Powerful numbers (A001694) which can be written as the sum of two relatively prime 3-powerful numbers (A036966) different from 1.

%I A210470 #37 Jan 30 2023 02:37:03
%S A210470 841,968,2312,3528,5041,5776,12769,14884,16641,45125,51984,109561,
%T A210470 123823,157609,168921,207576,373321,450241,498436,609725,711828,
%U A210470 731025,798768,940896,1223048,1590121,1792921,2478843,2481992,2526752,3157729,3964081,5346675,6255001
%N A210470 Powerful numbers (A001694) which can be written as the sum of two relatively prime 3-powerful numbers (A036966) different from 1.
%D A210470 Jean-Marie de Konninck, Those Fascinating Numbers, Amer. Math. Soc., 2009.
%D A210470 Alonso Del Arte, Posting to the Sequence Fans Mailing List, Mar 10 2011.
%H A210470 Amiram Eldar, <a href="/A210470/b210470.txt">Table of n, a(n) for n = 1..100</a>
%F A210470 { a in A001694: a=b+c and b,c >1 and b,c in A036966 and gcd(b,c)=1}. - _R. J. Mathar_, May 01 2013
%e A210470 841 = 216+625 ; 968 = 343+625 ; 2312=125+2187;
%p A210470 isA210470 := proc(n)
%p A210470     if isA001694(n) then
%p A210470         for i from 2 do
%p A210470             p3 := A036966(i) ;
%p A210470             if p3+2 > n then
%p A210470                 return false;
%p A210470             end if;
%p A210470             p3comp := n-p3 ;
%p A210470             if isA036966(p3comp) and igcd(p3,p3comp) = 1 then
%p A210470                 # print(n,p3,p3comp) ;
%p A210470                 return true;
%p A210470             end if;
%p A210470         end do:
%p A210470         return false;
%p A210470     else
%p A210470         return false;
%p A210470     end if;
%p A210470 end proc:
%p A210470 for n from 1 do
%p A210470     if isA210470(n) then
%p A210470         printf("%d,",n) ;
%p A210470     end if;
%p A210470 end do: # _R. J. Mathar_, May 01 2013
%t A210470 With[{max = 10^7}, powQ[n_, e_] := Min[FactorInteger[n][[;; , 2]]] > e; pows = Union[Flatten[Table[i^2*j^3, {j, max^(1/3)}, {i, Sqrt[max/j^3]}]]]; Select[Union[Plus @@@ Select[Tuples[Select[pows, powQ[#, 2] &], {2}], CoprimeQ @@ # &]], # < max && powQ[#, 1] &]] (* _Amiram Eldar_, Jan 30 2023 *)
%Y A210470 Cf. A001694, A036966.
%K A210470 nonn
%O A210470 1,1
%A A210470 _N. J. A. Sloane_, Apr 22 2013
%E A210470 More terms from _Amiram Eldar_, Jan 30 2023