A figurate number of form n(2n-1). more
The first 600 hexagonal numbers :
1,
6,
15,
28,
45,
66,
91,
120,
153,
190,
231,
276,
325,
378,
435,
496,
561,
630,
703,
780,
861,
946,
1035,
1128,
1225,
1326,
1431,
1540,
1653,
1770,
1891,
2016,
2145,
2278,
2415,
2556,
2701,
2850,
3003,
3160,
3321,
3486,
3655,
3828,
4005,
4186,
4371,
4560,
4753,
4950,
5151,
5356,
5565,
5778,
5995,
6216,
6441,
6670,
6903,
7140,
7381,
7626,
7875,
8128,
8385,
8646,
8911,
9180,
9453,
9730,
10011,
10296,
10585,
10878,
11175,
11476,
11781,
12090,
12403,
12720,
13041,
13366,
13695,
14028,
14365,
14706,
15051,
15400,
15753,
16110,
16471,
16836,
17205,
17578,
17955,
18336,
18721,
19110,
19503,
19900,
20301,
20706,
21115,
21528,
21945,
22366,
22791,
23220,
23653,
24090,
24531,
24976,
25425,
25878,
26335,
26796,
27261,
27730,
28203,
28680,
29161,
29646,
30135,
30628,
31125,
31626,
32131,
32640,
33153,
33670,
34191,
34716,
35245,
35778,
36315,
36856,
37401,
37950,
38503,
39060,
39621,
40186,
40755,
41328,
41905,
42486,
43071,
43660,
44253,
44850,
45451,
46056,
46665,
47278,
47895,
48516,
49141,
49770,
50403,
51040,
51681,
52326,
52975,
53628,
54285,
54946,
55611,
56280,
56953,
57630,
58311,
58996,
59685,
60378,
61075,
61776,
62481,
63190,
63903,
64620,
65341,
66066,
66795,
67528,
68265,
69006,
69751,
70500,
71253,
72010,
72771,
73536,
74305,
75078,
75855,
76636,
77421,
78210,
79003,
79800,
80601,
81406,
82215,
83028,
83845,
84666,
85491,
86320,
87153,
87990,
88831,
89676,
90525,
91378,
92235,
93096,
93961,
94830,
95703,
96580,
97461,
98346,
99235,
100128,
101025,
101926,
102831,
103740,
104653,
105570,
106491,
107416,
108345,
109278,
110215,
111156,
112101,
113050,
114003,
114960,
115921,
116886,
117855,
118828,
119805,
120786,
121771,
122760,
123753,
124750,
125751,
126756,
127765,
128778,
129795,
130816,
131841,
132870,
133903,
134940,
135981,
137026,
138075,
139128,
140185,
141246,
142311,
143380,
144453,
145530,
146611,
147696,
148785,
149878,
150975,
152076,
153181,
154290,
155403,
156520,
157641,
158766,
159895,
161028,
162165,
163306,
164451,
165600,
166753,
167910,
169071,
170236,
171405,
172578,
173755,
174936,
176121,
177310,
178503,
179700,
180901,
182106,
183315,
184528,
185745,
186966,
188191,
189420,
190653,
191890,
193131,
194376,
195625,
196878,
198135,
199396,
200661,
201930,
203203,
204480,
205761,
207046,
208335,
209628,
210925,
212226,
213531,
214840,
216153,
217470,
218791,
220116,
221445,
222778,
224115,
225456,
226801,
228150,
229503,
230860,
232221,
233586,
234955,
236328,
237705,
239086,
240471,
241860,
243253,
244650,
246051,
247456,
248865,
250278,
251695,
253116,
254541,
255970,
257403,
258840,
260281,
261726,
263175,
264628,
266085,
267546,
269011,
270480,
271953,
273430,
274911,
276396,
277885,
279378,
280875,
282376,
283881,
285390,
286903,
288420,
289941,
291466,
292995,
294528,
296065,
297606,
299151,
300700,
302253,
303810,
305371,
306936,
308505,
310078,
311655,
313236,
314821,
316410,
318003,
319600,
321201,
322806,
324415,
326028,
327645,
329266,
330891,
332520,
334153,
335790,
337431,
339076,
340725,
342378,
344035,
345696,
347361,
349030,
350703,
352380,
354061,
355746,
357435,
359128,
360825,
362526,
364231,
365940,
367653,
369370,
371091,
372816,
374545,
376278,
378015,
379756,
381501,
383250,
385003,
386760,
388521,
390286,
392055,
393828,
395605,
397386,
399171,
400960,
402753,
404550,
406351,
408156,
409965,
411778,
413595,
415416,
417241,
419070,
420903,
422740,
424581,
426426,
428275,
430128,
431985,
433846,
435711,
437580,
439453,
441330,
443211,
445096,
446985,
448878,
450775,
452676,
454581,
456490,
458403,
460320,
462241,
464166,
466095,
468028,
469965,
471906,
473851,
475800,
477753,
479710,
481671,
483636,
485605,
487578,
489555,
491536,
493521,
495510,
497503,
499500,
501501,
503506,
505515,
507528,
509545,
511566,
513591,
515620,
517653,
519690,
521731,
523776,
525825,
527878,
529935,
531996,
534061,
536130,
538203,
540280,
542361,
544446,
546535,
548628,
550725,
552826,
554931,
557040,
559153,
561270,
563391,
565516,
567645,
569778,
571915,
574056,
576201,
578350,
580503,
582660,
584821,
586986,
589155,
591328,
593505,
595686,
597871,
600060,
602253,
604450,
606651,
608856,
611065,
613278,
615495,
617716,
619941,
622170,
624403,
626640,
628881,
631126,
633375,
635628,
637885,
640146,
642411,
644680,
646953,
649230,
651511,
653796,
656085,
658378,
660675,
662976,
665281,
667590,
669903,
672220,
674541,
676866,
679195,
681528,
683865,
686206,
688551,
690900,
693253,
695610,
697971,
700336,
702705,
705078,
707455,
709836,
712221,
714610,
717003,
719400.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 22360680 values, from 1 to 999999997764120).
n\r | 0 | 1 |
2 | 11180340 | 11180340 | 2 |
3 | 14907120 | 7453560 | 0 | 3 |
4 | 5590170 | 5590170 | 5590170 | 5590170 | 4 |
5 | 8944272 | 8944272 | 0 | 4472136 | 0 | 5 |
6 | 7453560 | 3726780 | 0 | 7453560 | 3726780 | 0 | 6 |
7 | 6388765 | 6388766 | 0 | 6388766 | 0 | 0 | 3194383 | 7 |
8 | 2795085 | 2795085 | 2795085 | 2795085 | 2795085 | 2795085 | 2795085 | 2795085 | 8 |
9 | 4969040 | 7453560 | 0 | 4969040 | 0 | 0 | 4969040 | 0 | 0 | 9 |
10 | 4472136 | 4472136 | 0 | 2236068 | 0 | 4472136 | 4472136 | 0 | 2236068 | 0 | 10 |
11 | 4065578 | 4065579 | 0 | 4065578 | 2032789 | 0 | 4065578 | 0 | 0 | 0 | 4065578 |
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.