A figurate number of form n(3n-2). more
The first 600 octagonal numbers :
1,
8,
21,
40,
65,
96,
133,
176,
225,
280,
341,
408,
481,
560,
645,
736,
833,
936,
1045,
1160,
1281,
1408,
1541,
1680,
1825,
1976,
2133,
2296,
2465,
2640,
2821,
3008,
3201,
3400,
3605,
3816,
4033,
4256,
4485,
4720,
4961,
5208,
5461,
5720,
5985,
6256,
6533,
6816,
7105,
7400,
7701,
8008,
8321,
8640,
8965,
9296,
9633,
9976,
10325,
10680,
11041,
11408,
11781,
12160,
12545,
12936,
13333,
13736,
14145,
14560,
14981,
15408,
15841,
16280,
16725,
17176,
17633,
18096,
18565,
19040,
19521,
20008,
20501,
21000,
21505,
22016,
22533,
23056,
23585,
24120,
24661,
25208,
25761,
26320,
26885,
27456,
28033,
28616,
29205,
29800,
30401,
31008,
31621,
32240,
32865,
33496,
34133,
34776,
35425,
36080,
36741,
37408,
38081,
38760,
39445,
40136,
40833,
41536,
42245,
42960,
43681,
44408,
45141,
45880,
46625,
47376,
48133,
48896,
49665,
50440,
51221,
52008,
52801,
53600,
54405,
55216,
56033,
56856,
57685,
58520,
59361,
60208,
61061,
61920,
62785,
63656,
64533,
65416,
66305,
67200,
68101,
69008,
69921,
70840,
71765,
72696,
73633,
74576,
75525,
76480,
77441,
78408,
79381,
80360,
81345,
82336,
83333,
84336,
85345,
86360,
87381,
88408,
89441,
90480,
91525,
92576,
93633,
94696,
95765,
96840,
97921,
99008,
100101,
101200,
102305,
103416,
104533,
105656,
106785,
107920,
109061,
110208,
111361,
112520,
113685,
114856,
116033,
117216,
118405,
119600,
120801,
122008,
123221,
124440,
125665,
126896,
128133,
129376,
130625,
131880,
133141,
134408,
135681,
136960,
138245,
139536,
140833,
142136,
143445,
144760,
146081,
147408,
148741,
150080,
151425,
152776,
154133,
155496,
156865,
158240,
159621,
161008,
162401,
163800,
165205,
166616,
168033,
169456,
170885,
172320,
173761,
175208,
176661,
178120,
179585,
181056,
182533,
184016,
185505,
187000,
188501,
190008,
191521,
193040,
194565,
196096,
197633,
199176,
200725,
202280,
203841,
205408,
206981,
208560,
210145,
211736,
213333,
214936,
216545,
218160,
219781,
221408,
223041,
224680,
226325,
227976,
229633,
231296,
232965,
234640,
236321,
238008,
239701,
241400,
243105,
244816,
246533,
248256,
249985,
251720,
253461,
255208,
256961,
258720,
260485,
262256,
264033,
265816,
267605,
269400,
271201,
273008,
274821,
276640,
278465,
280296,
282133,
283976,
285825,
287680,
289541,
291408,
293281,
295160,
297045,
298936,
300833,
302736,
304645,
306560,
308481,
310408,
312341,
314280,
316225,
318176,
320133,
322096,
324065,
326040,
328021,
330008,
332001,
334000,
336005,
338016,
340033,
342056,
344085,
346120,
348161,
350208,
352261,
354320,
356385,
358456,
360533,
362616,
364705,
366800,
368901,
371008,
373121,
375240,
377365,
379496,
381633,
383776,
385925,
388080,
390241,
392408,
394581,
396760,
398945,
401136,
403333,
405536,
407745,
409960,
412181,
414408,
416641,
418880,
421125,
423376,
425633,
427896,
430165,
432440,
434721,
437008,
439301,
441600,
443905,
446216,
448533,
450856,
453185,
455520,
457861,
460208,
462561,
464920,
467285,
469656,
472033,
474416,
476805,
479200,
481601,
484008,
486421,
488840,
491265,
493696,
496133,
498576,
501025,
503480,
505941,
508408,
510881,
513360,
515845,
518336,
520833,
523336,
525845,
528360,
530881,
533408,
535941,
538480,
541025,
543576,
546133,
548696,
551265,
553840,
556421,
559008,
561601,
564200,
566805,
569416,
572033,
574656,
577285,
579920,
582561,
585208,
587861,
590520,
593185,
595856,
598533,
601216,
603905,
606600,
609301,
612008,
614721,
617440,
620165,
622896,
625633,
628376,
631125,
633880,
636641,
639408,
642181,
644960,
647745,
650536,
653333,
656136,
658945,
661760,
664581,
667408,
670241,
673080,
675925,
678776,
681633,
684496,
687365,
690240,
693121,
696008,
698901,
701800,
704705,
707616,
710533,
713456,
716385,
719320,
722261,
725208,
728161,
731120,
734085,
737056,
740033,
743016,
746005,
749000,
752001,
755008,
758021,
761040,
764065,
767096,
770133,
773176,
776225,
779280,
782341,
785408,
788481,
791560,
794645,
797736,
800833,
803936,
807045,
810160,
813281,
816408,
819541,
822680,
825825,
828976,
832133,
835296,
838465,
841640,
844821,
848008,
851201,
854400,
857605,
860816,
864033,
867256,
870485,
873720,
876961,
880208,
883461,
886720,
889985,
893256,
896533,
899816,
903105,
906400,
909701,
913008,
916321,
919640,
922965,
926296,
929633,
932976,
936325,
939680,
943041,
946408,
949781,
953160,
956545,
959936,
963333,
966736,
970145,
973560,
976981,
980408,
983841,
987280,
990725,
994176,
997633,
1001096,
1004565,
1008040,
1011521,
1015008,
1018501,
1022000,
1025505,
1029016,
1032533,
1036056,
1039585,
1043120,
1046661,
1050208,
1053761,
1057320,
1060885,
1064456,
1068033,
1071616,
1075205,
1078800.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 18257418 values, from 1 to 999999899565336).
n\r | 0 | 1 |
2 | 9128709 | 9128709 | 2 |
3 | 6085806 | 6085806 | 6085806 | 3 |
4 | 9128709 | 9128709 | 0 | 0 | 4 |
5 | 7302966 | 7302968 | 0 | 3651484 | 0 | 5 |
6 | 3042903 | 3042903 | 3042903 | 3042903 | 3042903 | 3042903 | 6 |
7 | 5216405 | 5216406 | 2608202 | 0 | 0 | 5216405 | 0 | 7 |
8 | 9128709 | 4564355 | 0 | 0 | 0 | 4564354 | 0 | 0 | 8 |
9 | 2028602 | 2028602 | 2028602 | 2028602 | 2028602 | 2028602 | 2028602 | 2028602 | 2028602 | 9 |
10 | 3651483 | 3651484 | 0 | 1825742 | 0 | 3651483 | 3651484 | 0 | 1825742 | 0 | 10 |
11 | 3319530 | 3319531 | 0 | 0 | 0 | 3319530 | 0 | 1659765 | 3319531 | 0 | 3319531 |
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.