A055394 Numbers that are the sum of a positive square and a positive cube.

2, 5, 9, 10, 12, 17, 24, 26, 28, 31, 33, 36, 37, 43, 44, 50, 52, 57, 63, 65, 68, 72, 73, 76, 80, 82, 89, 91, 100, 101, 108, 113, 122, 126, 127, 128, 129, 134, 141, 145, 148, 150, 152, 161, 164, 170, 171, 174, 177, 185, 189, 196, 197, 204, 206, 208, 217, 220, 223

Offset: 1

Author

Henry Bottomley, May 12 2000

Keywords

• easy
• nonn

Comments

This sequence was the subject of a question in the German mathematics competition Bundeswettbewerb Mathematik 2017 (see links). The second round contained a question A4 which asks readers to "Show that there are an infinite number of a such that a-1, a, and a+1 are members of A055394". - N. J. A. Sloane, Jul 04 2017 and Oct 14 2017

This sequence was also the subject of a question in the 22nd All-Russian Mathematical Olympiad 1996 (see link). The 1st question of the final round for Grade 9 asked competitors "What numbers are more frequent among the integers from 1 to 1000000: those that can be written as a sum of a square and a positive cube, or those that cannot be?" Answer is that there are more numbers not of this form. - Bernard Schott, Feb 18 2022

Examples

			a(5)=17 since 17=3^2+2^3.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Bundeswettbewerb Mathematik 2017, Der Wettbewerb in der 47 Runde
• Bundeswettbewerb Mathematik 2017, Aufgaben und Lösungen
• The IMO Compendium, Problem 1, 22nd All-Russian Mathematical Olympiad 1996.
• Index to sequences related to Olympiads and other Mathematical competitions.

Crossrefs

Cf. A022549, A055393, A078360. Complement of A066650.

Programs

Maple

isA055394 := proc(n)
    local a,b;
    for b from 1 do
        if b^3 >= n then
            return false;
        end if;
        asqr := n-b^3 ;
        if asqr >= 0 and issqr(asqr) then
            return true;
        end if;
    end do:
    return;
end proc:
for n from 1 to 1000 do
    if isA055394(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 03 2015

Mathematica

r[n_, y_] := Reduce[x > 0 && n == x^2 + y^3, x, Integers]; ok[n_] := Catch[Do[If[r[n, y] =!= False, Throw[True]], {y, 1, Ceiling[n^(1/3)]}]] == True; Select[Range[300], ok] (* Jean-François Alcover, Jul 16 2012 *)
solQ[n_] := Length[Reduce[p^2 + q^3 == n && p > 0 && q > 0, {p, q}, Integers]] > 0; Select[Range[224], solQ] (* Jayanta Basu, Jul 11 2013 *)
isQ[n_] := For[k = 1, k <= (n-1)^(1/3), k++, If[IntegerQ[Sqrt[n-k^3]], Return[True]]; False];
Select[Range[1000], isQ] (* Jean-François Alcover, Apr 06 2021, after Charles R Greathouse IV *)

PARI

list(lim)=my(v=List()); for(n=1,sqrtint(lim\1-1), for(m=1,sqrtnint(lim\1-n^2,3), listput(v,n^2+m^3))); Set(v) \\ Charles R Greathouse IV, May 15 2015

PARI

is(n)=for(k=1,sqrtnint(n-1,3), if(issquare(n-k^3), return(1))); 0 \\ Charles R Greathouse IV, May 15 2015

Formula

a(n) >> n^(6/5). - Charles R Greathouse IV, May 15 2015