%I #77 Jul 18 2016 21:26:07

%S 4624776,15438250,27445392,36998208,123506000,127396178,216226981,

%T 219563136,238328064,269442621,295985664,310289733,406767816,

%U 423432360,449519625,510200217,578097000,590421637,632767581,634207392,715674609,751462677

%N Taxi-cab numbers (A001235) that are the sum of two nonzero squares in more than one way.

%C Motivation was that question: What are the numbers that are the sums of 2 positive cubes in more than 1 way and also sums of 2 positive squares in more than 1 way?

%C A001235(99) = 4624776 = 2^3*3^6*13*61 is the least number with this property.

%C A taxi-cab number (A001235) can be the sum of two nonzero squares in exactly one way. For example 22754277 is the least taxi-cab number that is the sum of two nonzero squares in exactly one way. 22754277 = 69^3 + 282^3 = 189^3 + 252^3 = 2646^2 + 3969^2. So 22754277 is not a member of this sequence. The next one is 8*22754277 = 182034216 = 138^3 + 564^3 = 378^3 + 504^3 = 2646^2 + 13230^2.

%C A taxi-cab number (A001235) can be of the form 2*n^2. For example 760032072 is the least number with this property. 760032072 = 114^3 + 912^3 = 513^3 + 855^3 = 2*19494^2. Note that 760032072 is a term of A081324. So it is not a term of this sequence.

%C 216226981 = 373*661*877 is the first term that has three prime divisors. It is also first squarefree term in this sequence.

%C It is easy to see that this sequence is infinite.

%H Chai Wah Wu, <a href="/A272701/b272701.txt">Table of n, a(n) for n = 1..643</a>

%e 4624776 = 51^3 + 165^3 = 72^3 + 162^3 = 1026^2 + 1890^2 = 1350^2 + 1674^2.

%e 27445392 = 141^3 + 291^3 = 198^3 + 270^3 = 756^2 + 5184^2 = 1296^2 + 5076^2.

%e 36998208 = 102^3 + 330^3 = 144^3 + 324^3 = 648^2 + 6048^2 = 1728^2 + 5832^2.

%o (PARI) T = thueinit(x^3+1, 1);

%o isA001235(n) = {my(v=thue(T, n)); sum(i=1, #v, v[i][1]>=0 && v[i][2]>=v[i][1])>1;}

%o isA007692(n) = {nb = 0; lim = sqrtint(n); for (x=1, lim, if ((n-x^2 >= x^2) && issquare(n-x^2), nb++);); nb >= 2;}

%o isok(n) = isA001235(n) && isA007692(n);

%Y Cf. A001235, A007692.

%K nonn

%O 1,1

%A _Altug Alkan_, May 12 2016