A272701 Taxi-cab numbers (A001235) that are the sum of two nonzero squares in more than one way.

4624776, 15438250, 27445392, 36998208, 123506000, 127396178, 216226981, 219563136, 238328064, 269442621, 295985664, 310289733, 406767816, 423432360, 449519625, 510200217, 578097000, 590421637, 632767581, 634207392, 715674609, 751462677

Offset: 1

Author

Altug Alkan, May 12 2016

Keywords

• nonn

Comments

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?

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

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.

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.

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

It is easy to see that this sequence is infinite.

Examples

			4624776 = 51^3 + 165^3 = 72^3 + 162^3 = 1026^2 + 1890^2 = 1350^2 + 1674^2.
27445392 = 141^3 + 291^3 = 198^3 + 270^3 = 756^2 + 5184^2 = 1296^2 + 5076^2.
36998208 = 102^3 + 330^3 = 144^3 + 324^3 = 648^2 + 6048^2 = 1728^2 + 5832^2.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..643

Crossrefs

Cf. A001235, A007692.

Programs

PARI

T = thueinit(x^3+1, 1);
isA001235(n) = {my(v=thue(T, n)); sum(i=1, #v, v[i][1]>=0 && v[i][2]>=v[i][1])>1;}
isA007692(n) = {nb = 0; lim = sqrtint(n); for (x=1, lim, if ((n-x^2 >= x^2) && issquare(n-x^2), nb++);); nb >= 2;}
isok(n) = isA001235(n) && isA007692(n);