A number that can be written as a sum of two positive cubes in more than one way. more
The first 600 taxicab numbers :
1729,
4104,
13832,
20683,
32832,
39312,
40033,
46683,
64232,
65728,
110656,
110808,
134379,
149389,
165464,
171288,
195841,
216027,
216125,
262656,
314496,
320264,
327763,
373464,
402597,
439101,
443889,
513000,
513856,
515375,
525824,
558441,
593047,
684019,
704977,
805688,
842751,
885248,
886464,
920673,
955016,
984067,
994688,
1009736,
1016496,
1061424,
1073375,
1075032,
1080891,
1092728,
1195112,
1260441,
1323712,
1331064,
1370304,
1407672,
1533357,
1566728,
1609272,
1728216,
1729000,
1734264,
1774656,
1845649,
2048391,
2101248,
2301299,
2418271,
2515968,
2562112,
2585375,
2622104,
2691451,
2864288,
2987712,
2991816,
3220776,
3242197,
3375001,
3375008,
3511872,
3512808,
3551112,
3587409,
3628233,
3798613,
3813992,
4033503,
4104000,
4110848,
4123000,
4174281,
4206592,
4342914,
4467528,
4505949,
4511808,
4607064,
4624776,
4673088,
4744376,
4914000,
4931101,
5004125,
5135104,
5287707,
5318677,
5446359,
5462424,
5472152,
5639816,
5772403,
5799339,
5832729,
5835375,
6013000,
6017193,
6058747,
6230504,
6434883,
6445504,
6517000,
6742008,
7081984,
7091712,
7094269,
7365384,
7620661,
7640128,
7872536,
7882245,
7957504,
8029000,
8077888,
8131968,
8216000,
8275176,
8491392,
8493039,
8494577,
8587000,
8600256,
8647128,
8741824,
8849601,
8872487,
9016488,
9261729,
9443761,
9560896,
10036845,
10083528,
10202696,
10424232,
10469816,
10589696,
10648512,
10702783,
10765603,
10811017,
10821896,
10870119,
10962432,
11097567,
11261376,
11478584,
11548089,
11557000,
11855727,
11859211,
11985003,
12266856,
12533824,
12650337,
12753160,
12874176,
13484016,
13489875,
13490533,
13623913,
13731319,
13825728,
13832000,
13851000,
13874112,
13915125,
14197248,
14482503,
14765192,
14916727,
14978061,
15077907,
15187041,
15201368,
15398936,
15438250,
15682680,
16012269,
16387128,
16387189,
16587207,
16776487,
16797375,
16809984,
16983854,
17045567,
17999072,
18406603,
18410392,
18468513,
18617625,
18673625,
19034379,
19250163,
19346168,
20072017,
20127744,
20162952,
20496896,
20616463,
20683000,
20976832,
21036743,
21125384,
21411000,
21531608,
21753576,
21994875,
22031576,
22426369,
22543417,
22544704,
22754277,
22914304,
23901696,
23934528,
24480125,
24666560,
24767171,
24858171,
24897817,
25164405,
25371171,
25766208,
25785432,
25937576,
26122131,
26506467,
26569809,
26856576,
27000008,
27000064,
27003375,
27015625,
27262872,
27445392,
27529073,
27831951,
27882127,
28094976,
28102464,
28149336,
28408896,
28658448,
28699272,
28703269,
28981125,
29025864,
29184057,
29503656,
29791125,
30388904,
30511936,
31256000,
32114143,
32268024,
32530680,
32832000,
32886784,
32973759,
32984000,
33108264,
33394248,
33652736,
34031907,
34743312,
35740224,
35938728,
36047592,
36050904,
36094464,
36856512,
36998208,
37384704,
37498032,
37955008,
38007144,
38253878,
39312000,
39448808,
39639691,
40033000,
40134584,
40183262,
40354119,
40519115,
40970375,
41080832,
41258737,
41301953,
41343640,
41400639,
41436873,
41511232,
41892515,
42154840,
42168581,
42276843,
42301656,
42549416,
43450344,
43570872,
43699392,
43777216,
43852536,
43946000,
43997625,
44409995,
45118528,
45440551,
45882739,
46091997,
46179224,
46343059,
46394712,
46661832,
46683000,
46825128,
47915712,
48104000,
48137544,
48316662,
48469976,
49832523,
49844032,
49849856,
49933368,
50324625,
51240427,
51278929,
51479064,
51508639,
51564032,
52136000,
52242463,
52324272,
52739288,
53226297,
53283923,
53936064,
54267003,
54269091,
54814509,
54887625,
55306557,
55486125,
56388339,
56655872,
56733696,
56754152,
57066633,
58438072,
58751784,
58923072,
59913728,
60271939,
60698521,
60965288,
61121024,
61555832,
61568208,
62108397,
62135073,
62980288,
63057960,
63660032,
64125000,
64232000,
64421875,
64623104,
65055744,
65272753,
65293317,
65573928,
65728000,
66201408,
66469429,
66763333,
67173463,
67931136,
67944312,
67956616,
68007673,
68696000,
68802048,
69177024,
69583995,
69625969,
69805125,
69934592,
70449093,
70796808,
70957971,
70979896,
71292312,
71509312,
72131904,
72669177,
73369088,
73819863,
73842713,
74093832,
74097261,
74130875,
75550088,
76487168,
77129864,
77335776,
77414967,
79598125,
79692193,
80102574,
80208576,
80294760,
80668224,
80779032,
81621568,
83393856,
83758528,
84266000,
84637287,
84717568,
85188096,
85492792,
85502375,
85622264,
86124824,
86316741,
86368464,
86488136,
86575168,
86960952,
87029432,
87483968,
87539319,
87579037,
87699456,
87952501,
88122125,
88129323,
88780536,
89576767,
90091008,
90527229,
91034307,
91125027,
91125216,
91140435,
91828672,
92384712,
92456000,
94100426,
94820544,
94845816,
94873688,
95547816,
95731489,
95880024,
96753187,
96860043,
97867441,
97962291,
98040969,
98134848,
99768753,
100092456,
100270592,
100711000,
101194704,
101202696,
101615579,
101700872,
102025280,
102562551,
102977784,
102993408,
104212017,
105343875,
105997327,
106243219,
106496767,
107872128,
107919000,
107924264,
108904581,
108991304,
109850552,
110087432,
110605824,
110656000,
110808000,
110992896,
111321000,
111485673,
111906333,
112422709,
112705587,
113577984,
114103808,
115084125,
115860024,
115942104,
116215603,
116773741,
117132750,
117258678,
118121536,
118129375,
118974960,
119082880,
119164409,
119333816,
119377000,
119793457,
119824488,
120623256,
121287376,
121496328,
121610944,
121660623,
121818816,
122262264,
122810129,
123008375,
123191488,
123262120,
123506000,
124336000,
124390728,
124868952,
125461440,
126173376,
126217000,
126516061,
126906624,
127062000,
127396178,
127525671,
128098152,
128966383,
129333078,
129380329,
131097024,
131097512,
131126303,
131997229,
132678000,
132697656,
133139727,
134171875,
134211896,
134379000,
134479872,
135111375,
135870832,
136004184,
136364536,
136591000,
137467603,
137854899,
138090771,
138647808,
141117704,
141864697,
142098327,
142244648,
142768089,
143604279,
143992576,
144370863,
144404416,
145461771,
145908847,
146548376,
147051693,
147187369,
147252824,
147283136,
147485448,
147748104.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 870850 values, from 1729 to 999999259966294).
n\r | 0 | 1 |
2 | 558502 | 312348 | 2 |
3 | 489438 | 196536 | 184876 | 3 |
4 | 532732 | 156521 | 25770 | 155827 | 4 |
5 | 208643 | 165249 | 165548 | 165897 | 165513 | 5 |
6 | 312567 | 79511 | 128910 | 176871 | 117025 | 55966 | 6 |
7 | 552468 | 42862 | 116357 | 0 | 0 | 115974 | 43189 | 7 |
8 | 522613 | 78786 | 12877 | 77832 | 10119 | 77735 | 12893 | 77995 | 8 |
9 | 489438 | 166235 | 27944 | 0 | 0 | 0 | 0 | 30301 | 156932 | 9 |
10 | 131671 | 58616 | 106422 | 58599 | 106478 | 76972 | 106633 | 59126 | 107298 | 59035 | 10 |
11 | 78767 | 79166 | 79200 | 79274 | 79333 | 79187 | 79043 | 79172 | 79247 | 79158 | 79303 |
A pictorial representation of the table above
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.