A050804 Numbers n such that n^3 is the sum of two nonzero squares in exactly one way.

2, 8, 18, 32, 72, 98, 128, 162, 242, 288, 392, 512, 648, 722, 882, 968, 1058, 1152, 1458, 1568, 1922, 2048, 2178, 2592, 2888, 3528, 3698, 3872, 4232, 4418, 4608, 4802, 5832, 6272, 6498, 6962, 7688, 7938, 8192

Offset

1,1

Comments

m is a term if and only if m = 2^(2a_0+1)*p_1^(2a_1)*p_2^(2a_2)*...*p_k^(2a_k), where a_i are nonnegative integers and p_i are primes of the form 4k+3. - Chai Wah Wu, Feb 27 2016

m is a term if and only if for all odd q > 1, m^q is the sum of two nonzero squares in exactly one way. - Chai Wah Wu, Feb 28 2016

Numbers n such that n is the sum of two nonzero squares while n^2 is not. - Altug Alkan, Apr 11 2016

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Index entries for sequences related to sums of squares

Formula

n such that A084888(n) = 1.

Example

E.g. 32^3 = 128^2 + 128^2. Is there an example using different squares?
No: If n^3 has only one representation as n^3 = a^2+b^2 with 0<a<=b, then a=b. - Jonathan Vos Post, Feb 02 2011

Mathematica

ok[n_] := Length @ Cases[ PowersRepresentations[n^3, 2, 2], {_?Positive, _?Positive}] == 1; Select[Range[8200], ok] (* Jean-François Alcover, Apr 05 2011 *)

Prog

(Haskell)
a050804 n = a050804_list !! (n-1)
a050804_list = filter ((== 1) . a084888) [0..]
-- Reinhard Zumkeller, Jul 18 2012
(Python)
from sympy import factorint
A050804_list = [2*i for i in range(1, 10**6) if not any(p % 4 == 1 or factorint(i)[p] % 2 for p in factorint(i))] # Chai Wah Wu, Feb 27 2016

Crossrefs

Cf. A000404, A050803.

Cf. A081324.

Sequence in context: A361905 A336489 A051787 * A081324 A361903 A190787

Adjacent sequences: A050801 A050802 A050803 * A050805 A050806 A050807

Keyword

nonn,nice

Author

Patrick De Geest, Sep 15 1999

Extensions

More terms from Michel ten Voorde and Jud McCranie

Status

approved