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 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
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.
FORMULA
a(n) >> n^(6/5). - Charles R Greathouse IV, May 15 2015
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 := 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 *)
PROG
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Henry Bottomley, May 12 2000
STATUS
approved