Figurate numbers of the form 3n(n-1)/2 + 1. more
The first 600 centered triangular numbers :
1,
4,
10,
19,
31,
46,
64,
85,
109,
136,
166,
199,
235,
274,
316,
361,
409,
460,
514,
571,
631,
694,
760,
829,
901,
976,
1054,
1135,
1219,
1306,
1396,
1489,
1585,
1684,
1786,
1891,
1999,
2110,
2224,
2341,
2461,
2584,
2710,
2839,
2971,
3106,
3244,
3385,
3529,
3676,
3826,
3979,
4135,
4294,
4456,
4621,
4789,
4960,
5134,
5311,
5491,
5674,
5860,
6049,
6241,
6436,
6634,
6835,
7039,
7246,
7456,
7669,
7885,
8104,
8326,
8551,
8779,
9010,
9244,
9481,
9721,
9964,
10210,
10459,
10711,
10966,
11224,
11485,
11749,
12016,
12286,
12559,
12835,
13114,
13396,
13681,
13969,
14260,
14554,
14851,
15151,
15454,
15760,
16069,
16381,
16696,
17014,
17335,
17659,
17986,
18316,
18649,
18985,
19324,
19666,
20011,
20359,
20710,
21064,
21421,
21781,
22144,
22510,
22879,
23251,
23626,
24004,
24385,
24769,
25156,
25546,
25939,
26335,
26734,
27136,
27541,
27949,
28360,
28774,
29191,
29611,
30034,
30460,
30889,
31321,
31756,
32194,
32635,
33079,
33526,
33976,
34429,
34885,
35344,
35806,
36271,
36739,
37210,
37684,
38161,
38641,
39124,
39610,
40099,
40591,
41086,
41584,
42085,
42589,
43096,
43606,
44119,
44635,
45154,
45676,
46201,
46729,
47260,
47794,
48331,
48871,
49414,
49960,
50509,
51061,
51616,
52174,
52735,
53299,
53866,
54436,
55009,
55585,
56164,
56746,
57331,
57919,
58510,
59104,
59701,
60301,
60904,
61510,
62119,
62731,
63346,
63964,
64585,
65209,
65836,
66466,
67099,
67735,
68374,
69016,
69661,
70309,
70960,
71614,
72271,
72931,
73594,
74260,
74929,
75601,
76276,
76954,
77635,
78319,
79006,
79696,
80389,
81085,
81784,
82486,
83191,
83899,
84610,
85324,
86041,
86761,
87484,
88210,
88939,
89671,
90406,
91144,
91885,
92629,
93376,
94126,
94879,
95635,
96394,
97156,
97921,
98689,
99460,
100234,
101011,
101791,
102574,
103360,
104149,
104941,
105736,
106534,
107335,
108139,
108946,
109756,
110569,
111385,
112204,
113026,
113851,
114679,
115510,
116344,
117181,
118021,
118864,
119710,
120559,
121411,
122266,
123124,
123985,
124849,
125716,
126586,
127459,
128335,
129214,
130096,
130981,
131869,
132760,
133654,
134551,
135451,
136354,
137260,
138169,
139081,
139996,
140914,
141835,
142759,
143686,
144616,
145549,
146485,
147424,
148366,
149311,
150259,
151210,
152164,
153121,
154081,
155044,
156010,
156979,
157951,
158926,
159904,
160885,
161869,
162856,
163846,
164839,
165835,
166834,
167836,
168841,
169849,
170860,
171874,
172891,
173911,
174934,
175960,
176989,
178021,
179056,
180094,
181135,
182179,
183226,
184276,
185329,
186385,
187444,
188506,
189571,
190639,
191710,
192784,
193861,
194941,
196024,
197110,
198199,
199291,
200386,
201484,
202585,
203689,
204796,
205906,
207019,
208135,
209254,
210376,
211501,
212629,
213760,
214894,
216031,
217171,
218314,
219460,
220609,
221761,
222916,
224074,
225235,
226399,
227566,
228736,
229909,
231085,
232264,
233446,
234631,
235819,
237010,
238204,
239401,
240601,
241804,
243010,
244219,
245431,
246646,
247864,
249085,
250309,
251536,
252766,
253999,
255235,
256474,
257716,
258961,
260209,
261460,
262714,
263971,
265231,
266494,
267760,
269029,
270301,
271576,
272854,
274135,
275419,
276706,
277996,
279289,
280585,
281884,
283186,
284491,
285799,
287110,
288424,
289741,
291061,
292384,
293710,
295039,
296371,
297706,
299044,
300385,
301729,
303076,
304426,
305779,
307135,
308494,
309856,
311221,
312589,
313960,
315334,
316711,
318091,
319474,
320860,
322249,
323641,
325036,
326434,
327835,
329239,
330646,
332056,
333469,
334885,
336304,
337726,
339151,
340579,
342010,
343444,
344881,
346321,
347764,
349210,
350659,
352111,
353566,
355024,
356485,
357949,
359416,
360886,
362359,
363835,
365314,
366796,
368281,
369769,
371260,
372754,
374251,
375751,
377254,
378760,
380269,
381781,
383296,
384814,
386335,
387859,
389386,
390916,
392449,
393985,
395524,
397066,
398611,
400159,
401710,
403264,
404821,
406381,
407944,
409510,
411079,
412651,
414226,
415804,
417385,
418969,
420556,
422146,
423739,
425335,
426934,
428536,
430141,
431749,
433360,
434974,
436591,
438211,
439834,
441460,
443089,
444721,
446356,
447994,
449635,
451279,
452926,
454576,
456229,
457885,
459544,
461206,
462871,
464539,
466210,
467884,
469561,
471241,
472924,
474610,
476299,
477991,
479686,
481384,
483085,
484789,
486496,
488206,
489919,
491635,
493354,
495076,
496801,
498529,
500260,
501994,
503731,
505471,
507214,
508960,
510709,
512461,
514216,
515974,
517735,
519499,
521266,
523036,
524809,
526585,
528364,
530146,
531931,
533719,
535510,
537304,
539101.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 25819889 values, from 1 to 999999963228649).
n\r | 0 | 1 |
2 | 12909944 | 12909945 | 2 |
3 | 0 | 25819889 | 0 | 3 |
4 | 6454972 | 6454973 | 6454972 | 6454972 | 4 |
5 | 5163978 | 10327955 | 0 | 0 | 10327956 | 5 |
6 | 0 | 12909945 | 0 | 0 | 12909944 | 0 | 6 |
7 | 0 | 7377111 | 0 | 7377111 | 7377111 | 3688556 | 0 | 7 |
8 | 3227486 | 3227487 | 3227486 | 3227486 | 3227486 | 3227486 | 3227486 | 3227486 | 8 |
9 | 0 | 17213259 | 0 | 0 | 8606630 | 0 | 0 | 0 | 0 | 9 |
10 | 2581989 | 5163978 | 0 | 0 | 5163978 | 2581989 | 5163977 | 0 | 0 | 5163978 | 10 |
11 | 0 | 4694525 | 2347263 | 0 | 4694525 | 0 | 0 | 0 | 4694525 | 4694526 | 4694525 |
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.