A178130 Numbers whose fifth power is the sum of a positive square and cube: n^5 = x^2 + y^3 with x, y > 0.

8, 19, 24, 28, 32, 36, 75, 81, 88, 96, 136, 176, 224, 225, 250, 328, 369, 395, 432, 432, 468, 500, 512, 537, 648, 701, 710, 864, 980, 1000, 1078, 1089, 1125, 1216, 1225, 1296, 1440, 1536, 1620, 1734, 1764, 1792, 1800, 1944, 1944, 2000, 2028, 2048, 2178, 2304

Offset

1,1

Comments

If a number appears more than once, it can be expressed in as many ways as its occurrence.

Zeros are not allowed otherwise 8 would appear twice, since 8^5 = 0^2 + 32^3 = 104^2 + 28^3. - Michel Marcus, Feb 25 2013

Cubes are not allowed to be negative, otherwise there would be two 8 and two 32, since 8^5 = 192^2 - 16^3 and 32^5=36352^2 - 1088^3. - Giovanni Resta, Feb 25 2013

Sequence is infinite: if k is a term then k*m^6, for any m, is also a term. For example, {8, 19, 24, 28, 32, 36}*2^6 = {512, 1216, 1536, 1792, 2048, 2304}, or terms 1, 2, 3, 4, 5, and 6 multiplied by 2^6 = terms 23, 34, 39, 42, 48, and 50. - Zak Seidov, Jun 27 2013

Links

Table of n, a(n) for n=1..50.

Example

For n=4 a(4)=28 since 28^5 = 3912^2 + 124^3.
Multiple representation can happen like 3528^5 = 714208320^2 + 331632^3 = 464679936^2 + 691488^3.
432 is the smallest having multiple representation, since 432^5 = 3732480^2 + 10368^3 = 3359232^2 + 15552^3.

Maple

f:= proc(z) local z5, y, x, m;
   z5:= z^5; m:= 0;
   for y from 1 do
      x:= z5 - y^3;
      if x <= 0 then return z$m fi;
      if issqr(x) then m:= m+1 fi;
   od
end proc:
map(f, [$1..2500]); # Robert Israel, Nov 13 2016

Mathematica

Reap[Do[m5 = m^5; Do[x2 = m5 - y^3; If[x2 <= 0, Break[]]; If[IntegerQ@ Sqrt@ x2, Print[m]; Sow[m]], {y, m5}], {m, 2500}]][[2, 1]] (* Jean-François Alcover, Sep 14 2020, after PARI *)

Prog

(PARI) lista() = for (m=1, 2500, n = m^5; for (i=1, n, v = n - i^3; if (v <= 0, break); if (issquare(v), print1(m, ", ")))) \\ Michel Marcus, Feb 25 2013

Crossrefs

Cf. A227029.

Sequence in context: A290185 A374671 A274397 * A227029 A260004 A144171

Adjacent sequences: A178127 A178128 A178129 * A178131 A178132 A178133

Keyword

nonn

Author

Carmine Suriano, May 20 2010

Extensions

It seems that 1392 is wrong and should be omitted. - Zak Seidov, Jun 25 2010

Script also found that 1392 is wrong, so 1392 has been removed and 2304 has been appended instead. - Michel Marcus, Feb 27 2013

1444 removed and 1734 inserted by Robert Israel, Nov 13 2016

Status

approved