

A178130


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


1



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
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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]] (* JeanFrançois 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: A294581 A290185 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



