A001235 -id:A001235 - cp's OEIS frontend

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

Altug Alkan, May 12 2016

nonn

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.

			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.

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

Cf. A001235, A007692.

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);

A272935 Taxi-cab numbers (A001235) that are the product of exactly three (not necessarily distinct) primes.

Original entry on oeis.org

1729, 20683, 149389, 195841, 327763, 2418271, 6058747, 7620661, 9443761, 10765603, 13623913, 18406603, 32114143, 68007673, 105997327, 106243219, 166560193, 216226981, 446686147, 584504191, 813357253, 959281759, 1098597061, 1736913439, 2072769211, 2460483307

Offset: 1

Altug Alkan, May 11 2016

nonn

Note that the sum of two positive cubes cannot be prime except 2, obviously. Additionally, if the sum of two positive cubes is a semiprime, then, all corresponding semiprimes have a unique representation as a sum of two distinct positive cubes (see comment section of A085366). Since we know that 1729 is the first member of A001235 and it has three prime divisors, the minimum value of the number of prime divisors of a taxi-cab number must be three. This was the motivation of the definition of this sequence.

			Taxi-cab number 1729 is a term because 1729 = 7*13*19.
Taxi-cab number 20683 is a term because 20683 = 13*37*43.
Taxi-cab number 149389 is a term because 149389 = 31*61*79.

Chai Wah Wu, Table of n, a(n) for n = 1..131

Cf. A001235, A003325, A014612, A085366.

a(7)-a(26) from Chai Wah Wu, May 22 2016

A259836 Integers n where n^3 + (n+1)^3 is a Taxicab number A001235.

Original entry on oeis.org

9, 121, 235, 301, 1090, 1293, 1524, 3152, 8010, 15556, 15934, 19247, 20244, 21498, 24015, 25363, 25556, 45462, 57872, 63758, 80016, 93349, 94701, 101929, 113098, 119942, 132414, 143653, 167147, 186540, 192629, 229508, 246122, 247318, 292154, 307534, 322870

Offset: 1

David Rabahy, Jul 06 2015

nonn

			9^3 + 10^3 = 1729 = A001235(1), so 9 is in the sequence.

David Rabahy and Alois P. Heinz and Chai Wah Wu, Table of n, a(n) for n = 1..90 (first 38 terms from David Rabahy, next 12 terms from Alois P. Heinz)

Cf. A001235, A005898.

filter:= proc(n)
  local D, b, a, Q;
  D:= numtheory:-divisors(n);
  for b in D do
    a:= n/b;
    Q:= 12*b - 3*a^2;
    if Q > 9 and issqr(Q) and Q < 9*a^2 then return true fi
  od;
  false
end proc:
select(x -> filter(x^3 +(x+1)^3), [$1..100000]); # Robert Israel, Jul 07 2015

Select[Range[10000], Length[PowersRepresentations[#^3 + (# + 1)^3, 2, 3]]==2 &] (* Vincenzo Librandi, Jul 10 2015 *)

from _future_ import division
from gmpy2 import is_square
from sympy import divisors
A259836_list = []
for n in range(10000):
    m = n**3+(n+1)**3
    for x in divisors(m):
        x2 = x**2
        if x2 > m:
            break
        if x != (2*n+1) and m < x*x2 and is_square(12*m//x-3*x2):
            A259836_list.append(n)
            break # Chai Wah Wu, Jan 10 2016

A282873 Taxicab numbers (A001235) whose 4th power is the sum of two positive cubes in a nontrivial way.

Original entry on oeis.org

10202696, 29791125, 48137544, 70957971, 81621568, 238329000, 275472792, 288975141, 385100352, 387352719, 553514689, 567663768, 652972544, 692612137, 728274456, 1051871977, 1104726168, 1275337000, 1299713688, 1402390152, 1484204904, 1906632000, 2203782336, 2311801128

Offset: 1

Chai Wah Wu, Feb 24 2017

nonn

If n = a^3 + b^3, then n^4 has a trivial decomposition as a sum of 2 cubes: n^4 = (an)^3 + (bn)^3.

Chai Wah Wu, Table of n, a(n) for n = 1..530

Cf. A001235, A003325, A051387, A282872.

A001235 INTERSECT A051387.

A197719 Position of n-th taxi-cab number A001235(n) in the sequence A003325 of sums of two positive cubes.

Original entry on oeis.org

61, 110, 248, 328, 445, 499, 510, 561, 697, 708, 1001, 1004, 1145, 1226, 1309, 1342, 1470, 1563, 1565, 1785, 2012, 2042, 2065, 2259, 2372, 2515, 2540, 2795, 2800, 2806, 2840, 2958, 3076, 3390, 3448, 3779, 3896, 4022, 4031, 4135, 4235, 4320, 4345, 4396, 4412

Offset: 1

Zak Seidov, Oct 17 2011

nonn

			First taxi-cab number A001235(1)=1729 is A003325(61) hence a(1)=61; 2nd taxi-cab number A001235(2)=4104 is A003325(110) hence a(2)=110.

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A001235, A003325.

A001235(n) = A003325(a(n)).

A272897 Largest prime factor of n-th taxi-cab number A001235(n).

Original entry on oeis.org

19, 19, 19, 43, 19, 13, 43, 19, 37, 79, 19, 19, 79, 79, 43, 61, 79, 127, 19, 19, 13, 43, 109, 19, 37, 139, 43, 19, 37, 31, 79, 43, 19, 139, 127, 127, 13, 19, 19, 61, 103, 151, 409, 73, 181, 13, 277, 79, 43, 79, 79, 19, 43, 139, 61, 19, 61, 79, 103, 127, 19, 37, 79, 163, 79, 19, 19

Offset: 1

Altug Alkan, May 09 2016

nonn

There are two versions of "taxicab numbers" that are A001235 and A011541. This sequence focuses on the version A001235.
All terms of this sequence are members of this sequence infinitely many times. For example, in this sequence there are infinitely many times "277" and there are infinitely many times "17".
Is a(n) >= 13 for all n? This is true for n <= 10000. - Robert Israel, May 09 2016

			a(1) = A006530(A001235(1)) = A006530(1729) = 19.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001235, A006530.

FactorInteger[Select[Range[2*10^5], Length[PowersRepresentations[#, 2, 3]] > 1 &]][[All, -1, 1]] (* Michael De Vlieger, May 10 2016, after Harvey P. Dale at A001235 *)

a(n) = A006530(A001235(n)).

A272910 Numbers n such that (n-1)^3 + (n+1)^3 is a taxi-cab number (A001235).

Original entry on oeis.org

19, 32, 93, 124, 208, 243, 308, 395, 427, 471, 603, 672, 1057, 1568, 1892, 2181, 2223, 2587, 3040, 3049, 4037, 4336, 5232, 5556, 6196, 6305, 6643, 8288, 8748, 10161, 10185, 10612, 10985, 12352, 13741, 14807, 16021, 17568, 20352, 20653, 24080, 27216, 27867, 31113, 31869, 32032, 32500, 36593

Offset: 1

Altug Alkan, May 09 2016

nonn

Numbers n such that 2*n*(n^2+3) is a member of A001235.
19 and 3049 are the only prime numbers in this sequence for n < 10^5.
How is the graph of second differences of this sequence?

			19 is a term because 18^3 + 20^3 = 13832 = 2^3 + 24^3.

Chai Wah Wu, Table of n, a(n) for n = 1..300

Cf. A001235, A153976, A259836.

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;
lista(nn) = for(n=1, nn, if(isA001235(2*n*(n^2+3)), print1(n, ", ")));

A273219 Taxi-cab numbers (A001235) that are the sum of three positive cubes.

Original entry on oeis.org

216027, 262656, 515375, 1092728, 1331064, 1533357, 1728216, 1845649, 2101248, 2515968, 2562112, 2622104, 2864288, 3511872, 3551112, 4033503, 4123000, 4511808, 4607064, 5004125, 5462424, 5832729, 6017193, 7091712, 7882245, 8491392, 8493039, 8494577, 8741824

Offset: 1

Altug Alkan, May 18 2016

nonn

A001235(158) = 10702783 = 7*13*337*349 is the least squarefree term of this sequence.

			216027 is a term because 216027 = 3^3 + 60^3 = 22^3 + 59^3 = 11^3 + 42^3 + 52^3.
262656 is a term because 262656 = 8^3 + 64^3 = 36^3 + 60^3 = 15^3 + 42^3 + 57^3.
515375 is a term because 515375 = 15^3 + 80^3 = 54^3 + 71^3 = 30^3 + 62^3 + 63^3.

Chai Wah Wu, Table of n, a(n) for n = 1..16254

Cf. A001235, A003072.

A273555 a(n) = A001235(n) - floor(A001235(n)^(1/3))^3.

Original entry on oeis.org

1, 8, 8, 1000, 64, 8, 729, 27, 232, 1728, 64, 216, 1728, 512, 8000, 4913, 729, 27, 125, 512, 64, 5832, 13331, 216, 13580, 125, 4913, 1000, 1856, 3375, 13824, 7073, 343, 2547, 8, 1331, 12167, 512, 1728, 8000, 13824, 13768, 24389, 9736, 16496, 216, 12167, 13824, 19683, 1

Offset: 1

Altug Alkan, May 25 2016

nonn

Noncube terms of this sequence are 232, 13331, 13580, 1856, 7073, 2547, ...

How is the distribution of noncube terms in this sequence? See also A273592 that is related with this question.

Cf. A001235, A055400, A273592.

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;
lista(nn) = for(n=1, nn, if(isA001235(n), print1(n-sqrtnint(n,3)^3, ", ")));

a(n) = A055400(A001235(n)). - Michel Marcus, May 25 2016

A273753 Taxi-cab numbers (A001235) that are the average of two positive cubes in more than one way.

Original entry on oeis.org

6742008, 53936064, 182034216, 431488512, 842751000, 1456273728, 1581292125, 2312508744, 3451908096, 4914923832, 6742008000, 8973612648, 11395366632, 11650189824, 12650337000, 14812191576, 18500069952, 22754277000, 27615264768, 33123485304, 39319390656

Offset: 1

Altug Alkan, May 29 2016

nonn

Motivation for this sequence is that question: What is the least odd term of this sequence?

1581292125 = 3^6*5^3*7*37*67 is the least odd number that is the term of this sequence.

			6742008 is a term because 6742008 = 46^3 + 188^3 = 126^3 + 168^3 = (14^3 + 238^3)/2 = (105^3 + 231^3)/2.
53936064 is a term because 53936064 = 2^3*6742008.
1581292125 is a term because 1581292125 = 50^3 + 1165^3 = 540^3 + 1125^3 = (435^3 + 1455^3)/2 = (909^3 + 1341^3)/2.

Chai Wah Wu, Table of n, a(n) for n = 1..64

Cf. A001235, A191345.

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;
isok(n) = isA001235(n) && isA001235(2*n);

a(5)-a(21) from Giovanni Resta, Jun 01 2016

cp's OEIS Frontend

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

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A272935 Taxi-cab numbers (A001235) that are the product of exactly three (not necessarily distinct) primes.

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A259836 Integers n where n^3 + (n+1)^3 is a Taxicab number A001235.

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A282873 Taxicab numbers (A001235) whose 4th power is the sum of two positive cubes in a nontrivial way.

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A197719 Position of n-th taxi-cab number A001235(n) in the sequence A003325 of sums of two positive cubes.

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A272897 Largest prime factor of n-th taxi-cab number A001235(n).

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A272910 Numbers n such that (n-1)^3 + (n+1)^3 is a taxi-cab number (A001235).

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A273219 Taxi-cab numbers (A001235) that are the sum of three positive cubes.

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A273555 a(n) = A001235(n) - floor(A001235(n)^(1/3))^3.

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