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

3, 6, 10, 11, 13, 18, 21, 25, 26, 27, 28, 29, 32, 33, 34, 37, 38, 40, 42, 44, 45, 47, 49, 51, 52, 53, 58, 59, 60, 64, 66, 68, 69, 73, 74, 77, 79, 81, 83, 84, 86, 88, 89, 90, 91, 92, 93, 96, 98, 101, 102, 105, 107, 109, 112, 114, 116, 117, 118, 123, 124, 125

Offset: 1

Author

Jonathan Vos Post, Sep 25 2006

Keywords

• easy
• nonn

Examples

			a(1) = 3 = 1^4 + 1^3 + 1^2.
a(2) = 6 = 1^4 + 1^3 + 2^2.
a(3) = 10 = 1^4 + 2^3 + 1^2.
a(4) = 11 = 1^4 + 1^3 + 3^2.
a(5) = 13 = 1^4 + 2^3 + 2^2.
a(6) = 18 = 1^4 + 1^3 + 4^2 = 1^4 + 2^3 + 3^2 = 2^4 + 1^3 + 1^2.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000290 (squares), A000578 (cubes), A000583 (4th powers), A055394 (numbers that are the sum of a positive square and a positive cube).

Programs

Maple

isA123053 := proc(n)
    local x,y,z ;
    for x from 1 do
        if x^2 > n then
            return false;
        end if;
        for y from 1 do
            if x^2+y^3> n then
                break;
            end if;
            for z from 1 do
                if x^2+y^3+z^4 > n then
                    break;
                elif x^2+y^3+z^4 = n then
                    return true;
                end if;
            end do:
        end do:
    end do:
end proc:
n := 1 ;
for c from 0 to 10000 do
    if isA123053(c) then
        printf("%d %d\n",n,c) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Sep 07 2020

Mathematica

Select[ Union[ Total /@ Tuples[{Range[64]^2, Range[8]^4, Range[16]^3}]], # < 200 &] (* Giovanni Resta, Jun 12 2016 *)

Formula

{A000290 \ 0} + {A000578 \ 0} + {A000583}. {a^4 + b^3 + c^2 for a,b,c>0}.

Extensions

38, 86, and 93 added and 108 deleted by Giovanni Resta, Jun 12 2016