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Taxi-cab numbers ( A001235) that are the sum of two nonzero squares in more than one way.
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4
4624776, 15438250, 27445392, 36998208, 123506000, 127396178, 216226981, 219563136, 238328064, 269442621, 295985664, 310289733, 406767816, 423432360, 449519625, 510200217, 578097000, 590421637, 632767581, 634207392, 715674609, 751462677
COMMENTS
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.
Taxi-cab numbers ( A001235) that are the product of exactly three (not necessarily distinct) primes.
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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.
Integers n where n^3 + (n+1)^3 is a Taxicab number A001235.
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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.
Taxicab numbers ( A001235) whose 4th power is the sum of two positive cubes in a nontrivial way.
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10202696, 29791125, 48137544, 70957971, 81621568, 238329000, 275472792, 288975141, 385100352, 387352719, 553514689, 567663768, 652972544, 692612137, 728274456, 1051871977, 1104726168, 1275337000, 1299713688, 1402390152, 1484204904, 1906632000, 2203782336, 2311801128
Position of n-th taxi-cab number A001235(n) in the sequence A003325 of sums of two positive cubes.
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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).
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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.
Numbers n such that (n-1)^3 + (n+1)^3 is a taxi-cab number ( A001235).
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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.
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.
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
Taxi-cab numbers ( A001235) that are the average of two positive cubes in more than one way.
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1
6742008, 53936064, 182034216, 431488512, 842751000, 1456273728, 1581292125, 2312508744, 3451908096, 4914923832, 6742008000, 8973612648, 11395366632, 11650189824, 12650337000, 14812191576, 18500069952, 22754277000, 27615264768, 33123485304, 39319390656
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