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

%I A055394 #43 Nov 02 2023 08:39:31
%S A055394 2,5,9,10,12,17,24,26,28,31,33,36,37,43,44,50,52,57,63,65,68,72,73,76,
%T A055394 80,82,89,91,100,101,108,113,122,126,127,128,129,134,141,145,148,150,
%U A055394 152,161,164,170,171,174,177,185,189,196,197,204,206,208,217,220,223
%N A055394 Numbers that are the sum of a positive square and a positive cube.
%C A055394 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
%C A055394 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
%H A055394 Charles R Greathouse IV, <a href="/A055394/b055394.txt">Table of n, a(n) for n = 1..10000</a>
%H A055394 Bundeswettbewerb Mathematik 2017, <a href="https://www.mathe-wettbewerbe.de/bwm/bwm-wettbewerb-1">Der Wettbewerb in der 47 Runde</a>
%H A055394 Bundeswettbewerb Mathematik 2017, <a href="https://www.mathe-wettbewerbe.de/fileadmin/Mathe-Wettbewerbe/Bundeswettbewerb_Mathematik/Dokumente/Aufgaben_und_Loesungen_BWM/loes_17_2_e.pdf">Aufgaben und Lösungen</a>
%H A055394 The IMO Compendium, <a href="https://imomath.com/othercomp/Rus/RusMO96.pdf"> Problem 1</a>, 22nd All-Russian Mathematical Olympiad 1996.
%H A055394 <a href="/index/O#Olympiads">Index to sequences related to Olympiads and other Mathematical competitions</a>.
%F A055394 a(n) >> n^(6/5). - _Charles R Greathouse IV_, May 15 2015
%e A055394 a(5)=17 since 17=3^2+2^3.
%p A055394 isA055394 := proc(n)
%p A055394     local a,b;
%p A055394     for b from 1 do
%p A055394         if b^3 >= n then
%p A055394             return false;
%p A055394         end if;
%p A055394         asqr := n-b^3 ;
%p A055394         if asqr >= 0 and issqr(asqr) then
%p A055394             return true;
%p A055394         end if;
%p A055394     end do:
%p A055394     return;
%p A055394 end proc:
%p A055394 for n from 1 to 1000 do
%p A055394     if isA055394(n) then
%p A055394         printf("%d,",n) ;
%p A055394     end if;
%p A055394 end do: # _R. J. Mathar_, Dec 03 2015
%t A055394 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 *)
%t A055394 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 *)
%t A055394 isQ[n_] := For[k = 1, k <= (n-1)^(1/3), k++, If[IntegerQ[Sqrt[n-k^3]], Return[True]]; False];
%t A055394 Select[Range[1000], isQ] (* _Jean-François Alcover_, Apr 06 2021, after _Charles R Greathouse IV_ *)
%o A055394 (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
%o A055394 (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
%Y A055394 Cf. A022549, A055393, A078360. Complement of A066650.
%K A055394 easy,nonn
%O A055394 1,1
%A A055394 _Henry Bottomley_, May 12 2000