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A055394 Numbers that are the sum of a positive square and a positive cube. 40
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 (list; graph; refs; listen; history; text; internal format)
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.

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

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 *)

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

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

Sequence in context: A295567 A100530 A155469 * A078360 A114995 A047619

Adjacent sequences:  A055391 A055392 A055393 * A055395 A055396 A055397

KEYWORD

easy,nonn

AUTHOR

Henry Bottomley, May 12 2000

STATUS

approved

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Last modified February 22 19:17 EST 2018. Contains 299469 sequences. (Running on oeis4.)