

A055394


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


51



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
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OFFSET

1,1


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 a1, 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 AllRussian 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


LINKS

The IMO Compendium, Problem 1, 22nd AllRussian Mathematical Olympiad 1996.


FORMULA



EXAMPLE

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


MAPLE

isA055394 := proc(n)
local a, b;
for b from 1 do
if b^3 >= n then
return false;
end if;
asqr := nb^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;


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] (* JeanFrançois 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 <= (n1)^(1/3), k++, If[IntegerQ[Sqrt[nk^3]], Return[True]]; False];


PROG

(PARI) list(lim)=my(v=List()); for(n=1, sqrtint(lim\11), for(m=1, sqrtnint(lim\1n^2, 3), listput(v, n^2+m^3))); Set(v) \\ Charles R Greathouse IV, May 15 2015


CROSSREFS



KEYWORD

easy,nonn


AUTHOR



STATUS

approved



