A055393 -id:A055393 - cp's OEIS frontend

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

Henry Bottomley, May 12 2000

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

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

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.

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

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

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 *)

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

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

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

A022549 Sum of a square and a nonnegative cube.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 10, 12, 16, 17, 24, 25, 26, 27, 28, 31, 33, 36, 37, 43, 44, 49, 50, 52, 57, 63, 64, 65, 68, 72, 73, 76, 80, 81, 82, 89, 91, 100, 101, 108, 113, 121, 122, 125, 126, 127, 128, 129, 134, 141, 144, 145

Offset: 1

N. J. A. Sloane

It appears that there are no modular constraints on this sequence; i.e., every residue class of every integer has representatives here. - Franklin T. Adams-Watters, Dec 03 2009

A045634(a(n)) > 0. - Reinhard Zumkeller, Jul 17 2010

R. Zumkeller, Table of n, a(n) for n = 1..10000 - _Reinhard Zumkeller_, Jul 17 2010

Complement of A022550; A002760 and A179509 are subsequences.

Cf. A055394, A045634, A055393, A123291, A169618, A123364, A111925.

q=30; imax=q^2; Select[Union[Flatten[Table[x^2+y^3, {y,0,q^(2/3)}, {x,0,q}]]], #<=imax&] (* Vladimir Joseph Stephan Orlovsky, Apr 20 2011 *)

is(n)=for(k=0,sqrtnint(n,3), if(issquare(n-k^3), return(1))); 0 \\ Charles R Greathouse IV, Aug 24 2020

list(lim)=my(v=List(),t); for(k=0,sqrtnint(lim\=1,3), t=k^3; for(n=0,sqrtint(lim-t), listput(v,t+n^2))); Set(v) \\ Charles R Greathouse IV, Aug 24 2020

A054402 Numbers that are the sum of a positive square and a positive cube in more than one way.

Original entry on oeis.org

17, 65, 89, 108, 129, 145, 225, 233, 252, 297, 316, 388, 449, 464, 505, 537, 548, 577, 593, 633, 730, 737, 745, 792, 793, 801, 873, 1025, 1088, 1090, 1116, 1289, 1304, 1305, 1367, 1412, 1441, 1452, 1529, 1585, 1601

Offset: 1

Henry Bottomley, May 12 2000

			a(1)=17 since 17 = 3^2 + 2^3 = 4^2 + 1^3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A055393, A055394.

lst={};Do[Do[AppendTo[lst,n^2+m^3],{n,5!}],{m,5!}];lst=Sort[lst]; lst2={};Do[If[lst[[n]]==lst[[n+1]],AppendTo[lst2,lst[[n]]]],{n,Length[lst]-1}];lst2; Take[Union[lst2],123] (* Vladimir Joseph Stephan Orlovsky, Jul 17 2009 *)

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

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

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A022549 Sum of a square and a nonnegative cube.

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A054402 Numbers that are the sum of a positive square and a positive cube in more than one way.

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