A087209 Numbers k such that k^2 = x^3 + y^4 with positive integers x, y.

3, 29, 45, 76, 162, 192, 397, 405, 612, 650, 1025, 1098, 1275, 1856, 2064, 2160, 2187, 2880, 3420, 4864, 5499, 6831, 6875, 8775, 9072, 10368, 12288, 12348, 15000, 21096, 21141, 25408, 25920, 26811, 29079, 30600, 30758, 32805, 33957, 36875, 39168

Offset

1,1

Comments

There are an infinite number of solutions. Parametrizations are given at the Alpern link.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10006 (terms 1..380 from Robert Israel)

Dario Alpern, Sum of powers

Example

3 is a term because 3^2 = 2^3 + 1^4.
45 is a term because 45^2 = 9^3 + 6^4 = 729 + 1296 = 2025.

Maple

N:= 10^10: # for terms <= sqrt(N)
Res:= NULL:
for x from 1 while x^3 < N do
  for y from 1 do
    w:= x^3 + y^4;
  if w > N then break fi;
  if issqr(w) then Res:= Res, sqrt(w) fi
od od:
sort(convert({Res}, list)); # Robert Israel, Apr 30 2026

Crossrefs

Cf. A087210, solutions with gcd(n, x, y) = 1.

Sequence in context: A341210 A106979 A367102 * A165440 A072306 A303137

Adjacent sequences: A087206 A087207 A087208 * A087210 A087211 A087212

Keyword

nonn

Author

Hugo Pfoertner, Aug 26 2003

Status

approved