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Taxi-cab numbers ( A001235) that are the sum of two nonzero squares in more than one way.
+20
4
4624776, 15438250, 27445392, 36998208, 123506000, 127396178, 216226981, 219563136, 238328064, 269442621, 295985664, 310289733, 406767816, 423432360, 449519625, 510200217, 578097000, 590421637, 632767581, 634207392, 715674609, 751462677
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.
EXAMPLE
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.
PROG
(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);
Taxi-cab numbers ( A001235) that are the product of exactly three (not necessarily distinct) primes.
+20
3
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
COMMENTS
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.
EXAMPLE
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.
Integers n where n^3 + (n+1)^3 is a Taxicab number A001235.
+20
2
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
EXAMPLE
9^3 + 10^3 = 1729 = A001235(1), so 9 is in the sequence.
MAPLE
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
MATHEMATICA
Select[Range[10000], Length[PowersRepresentations[#^3 + (# + 1)^3, 2, 3]]==2 &] (* Vincenzo Librandi, Jul 10 2015 *)
PROG
(Python 3.x)
start = 9
end = 500000
print(start, end)
cubes = []
t = end**3+(end+1)**3
max = int(t**(1/3)+.5)
for i in range(0, max+1):
cubes.append(i**3)
for x in range(start, end):
t = cubes[x]+cubes[x+1]
for i in range(1, x):
z = t-cubes[i]
n = int(z**(1/3)+.5)
if cubes[n] == z:
print(x, x+1, i, n, '\a')
(Python)
from __future__ import division
from gmpy2 import is_square
from sympy import divisors
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):
Taxicab numbers ( A001235) whose 4th power is the sum of two positive cubes in a nontrivial way.
+20
2
10202696, 29791125, 48137544, 70957971, 81621568, 238329000, 275472792, 288975141, 385100352, 387352719, 553514689, 567663768, 652972544, 692612137, 728274456, 1051871977, 1104726168, 1275337000, 1299713688, 1402390152, 1484204904, 1906632000, 2203782336, 2311801128
COMMENTS
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.
Position of n-th taxi-cab number A001235(n) in the sequence A003325 of sums of two positive cubes.
+20
1
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
Largest prime factor of n-th taxi-cab number A001235(n).
+20
1
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
COMMENTS
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
Numbers n such that (n-1)^3 + (n+1)^3 is a taxi-cab number ( A001235).
+20
1
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
COMMENTS
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?
EXAMPLE
19 is a term because 18^3 + 20^3 = 13832 = 2^3 + 24^3.
PROG
(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;
lista(nn) = for(n=1, nn, if(isA001235(2*n*(n^2+3)), print1(n, ", ")));
Taxi-cab numbers ( A001235) that are the sum of three positive cubes.
+20
1
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
COMMENTS
A001235(158) = 10702783 = 7*13*337*349 is the least squarefree term of this sequence.
EXAMPLE
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.
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
COMMENTS
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.
PROG
(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;
lista(nn) = for(n=1, nn, if(isA001235(n), print1(n-sqrtnint(n, 3)^3, ", ")));
Taxi-cab numbers ( A001235) that are the average of two positive cubes in more than one way.
+20
1
6742008, 53936064, 182034216, 431488512, 842751000, 1456273728, 1581292125, 2312508744, 3451908096, 4914923832, 6742008000, 8973612648, 11395366632, 11650189824, 12650337000, 14812191576, 18500069952, 22754277000, 27615264768, 33123485304, 39319390656
COMMENTS
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.
EXAMPLE
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.
PROG
(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;
isok(n) = isA001235(n) && isA001235(2*n);
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