

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.


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 := 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;
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] (* 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 *)


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
(PARI) is(n)=for(k=1, sqrtnint(n1, 3), if(issquare(nk^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



