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A123053 Sum of a positive square, a positive cube and a positive fourth power.

%I A123053 #13 Sep 07 2020 05:58:37
%S A123053 3,6,10,11,13,18,21,25,26,27,28,29,32,33,34,37,38,40,42,44,45,47,49,
%T A123053 51,52,53,58,59,60,64,66,68,69,73,74,77,79,81,83,84,86,88,89,90,91,92,
%U A123053 93,96,98,101,102,105,107,109,112,114,116,117,118,123,124,125
%N A123053 Sum of a positive square, a positive cube and a positive fourth power.
%H A123053 Giovanni Resta, <a href="/A123053/b123053.txt">Table of n, a(n) for n = 1..10000</a>
%F A123053 {A000290 \ 0} + {A000578 \ 0} + {A000583}. {a^4 + b^3 + c^2 for a,b,c>0}.
%e A123053 a(1) = 3 = 1^4 + 1^3 + 1^2.
%e A123053 a(2) = 6 = 1^4 + 1^3 + 2^2.
%e A123053 a(3) = 10 = 1^4 + 2^3 + 1^2.
%e A123053 a(4) = 11 = 1^4 + 1^3 + 3^2.
%e A123053 a(5) = 13 = 1^4 + 2^3 + 2^2.
%e A123053 a(6) = 18 = 1^4 + 1^3 + 4^2 = 1^4 + 2^3 + 3^2 = 2^4 + 1^3 + 1^2.
%p A123053 isA123053 := proc(n)
%p A123053     local x,y,z ;
%p A123053     for x from 1 do
%p A123053         if x^2 > n then
%p A123053             return false;
%p A123053         end if;
%p A123053         for y from 1 do
%p A123053             if x^2+y^3> n then
%p A123053                 break;
%p A123053             end if;
%p A123053             for z from 1 do
%p A123053                 if x^2+y^3+z^4 > n then
%p A123053                     break;
%p A123053                 elif x^2+y^3+z^4 = n then
%p A123053                     return true;
%p A123053                 end if;
%p A123053             end do:
%p A123053         end do:
%p A123053     end do:
%p A123053 end proc:
%p A123053 n := 1 ;
%p A123053 for c from 0 to 10000 do
%p A123053     if isA123053(c) then
%p A123053         printf("%d %d\n",n,c) ;
%p A123053         n := n+1 ;
%p A123053     end if;
%p A123053 end do: # _R. J. Mathar_, Sep 07 2020
%t A123053 Select[ Union[ Total /@ Tuples[{Range[64]^2, Range[8]^4, Range[16]^3}]], # < 200 &] (* _Giovanni Resta_, Jun 12 2016 *)
%Y A123053 Cf. A000290 (squares), A000578 (cubes), A000583 (4th powers), A055394 (numbers that are the sum of a positive square and a positive cube).
%K A123053 easy,nonn
%O A123053 1,1
%A A123053 _Jonathan Vos Post_, Sep 25 2006
%E A123053 38, 86, and 93 added and 108 deleted by _Giovanni Resta_, Jun 12 2016