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

Author

Patrick De Geest, Sep 15 1999

Keywords

• nonn
• nice

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

Examples

			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
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A000404, A050803.

Cf. A081324.

Programs

Haskell

a050804 n = a050804_list !! (n-1)
a050804_list = filter ((== 1) . a084888) [0..]
-- Reinhard Zumkeller, Jul 18 2012

Mathematica

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

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

Formula

n such that A084888(n) = 1.

Extensions

More terms from Michel ten Voorde and Jud McCranie