Figurate numbers of the form 4n(n-1) + 1. more
The first 600 centered octagonal numbers :
1,
9,
25,
49,
81,
121,
169,
225,
289,
361,
441,
529,
625,
729,
841,
961,
1089,
1225,
1369,
1521,
1681,
1849,
2025,
2209,
2401,
2601,
2809,
3025,
3249,
3481,
3721,
3969,
4225,
4489,
4761,
5041,
5329,
5625,
5929,
6241,
6561,
6889,
7225,
7569,
7921,
8281,
8649,
9025,
9409,
9801,
10201,
10609,
11025,
11449,
11881,
12321,
12769,
13225,
13689,
14161,
14641,
15129,
15625,
16129,
16641,
17161,
17689,
18225,
18769,
19321,
19881,
20449,
21025,
21609,
22201,
22801,
23409,
24025,
24649,
25281,
25921,
26569,
27225,
27889,
28561,
29241,
29929,
30625,
31329,
32041,
32761,
33489,
34225,
34969,
35721,
36481,
37249,
38025,
38809,
39601,
40401,
41209,
42025,
42849,
43681,
44521,
45369,
46225,
47089,
47961,
48841,
49729,
50625,
51529,
52441,
53361,
54289,
55225,
56169,
57121,
58081,
59049,
60025,
61009,
62001,
63001,
64009,
65025,
66049,
67081,
68121,
69169,
70225,
71289,
72361,
73441,
74529,
75625,
76729,
77841,
78961,
80089,
81225,
82369,
83521,
84681,
85849,
87025,
88209,
89401,
90601,
91809,
93025,
94249,
95481,
96721,
97969,
99225,
100489,
101761,
103041,
104329,
105625,
106929,
108241,
109561,
110889,
112225,
113569,
114921,
116281,
117649,
119025,
120409,
121801,
123201,
124609,
126025,
127449,
128881,
130321,
131769,
133225,
134689,
136161,
137641,
139129,
140625,
142129,
143641,
145161,
146689,
148225,
149769,
151321,
152881,
154449,
156025,
157609,
159201,
160801,
162409,
164025,
165649,
167281,
168921,
170569,
172225,
173889,
175561,
177241,
178929,
180625,
182329,
184041,
185761,
187489,
189225,
190969,
192721,
194481,
196249,
198025,
199809,
201601,
203401,
205209,
207025,
208849,
210681,
212521,
214369,
216225,
218089,
219961,
221841,
223729,
225625,
227529,
229441,
231361,
233289,
235225,
237169,
239121,
241081,
243049,
245025,
247009,
249001,
251001,
253009,
255025,
257049,
259081,
261121,
263169,
265225,
267289,
269361,
271441,
273529,
275625,
277729,
279841,
281961,
284089,
286225,
288369,
290521,
292681,
294849,
297025,
299209,
301401,
303601,
305809,
308025,
310249,
312481,
314721,
316969,
319225,
321489,
323761,
326041,
328329,
330625,
332929,
335241,
337561,
339889,
342225,
344569,
346921,
349281,
351649,
354025,
356409,
358801,
361201,
363609,
366025,
368449,
370881,
373321,
375769,
378225,
380689,
383161,
385641,
388129,
390625,
393129,
395641,
398161,
400689,
403225,
405769,
408321,
410881,
413449,
416025,
418609,
421201,
423801,
426409,
429025,
431649,
434281,
436921,
439569,
442225,
444889,
447561,
450241,
452929,
455625,
458329,
461041,
463761,
466489,
469225,
471969,
474721,
477481,
480249,
483025,
485809,
488601,
491401,
494209,
497025,
499849,
502681,
505521,
508369,
511225,
514089,
516961,
519841,
522729,
525625,
528529,
531441,
534361,
537289,
540225,
543169,
546121,
549081,
552049,
555025,
558009,
561001,
564001,
567009,
570025,
573049,
576081,
579121,
582169,
585225,
588289,
591361,
594441,
597529,
600625,
603729,
606841,
609961,
613089,
616225,
619369,
622521,
625681,
628849,
632025,
635209,
638401,
641601,
644809,
648025,
651249,
654481,
657721,
660969,
664225,
667489,
670761,
674041,
677329,
680625,
683929,
687241,
690561,
693889,
697225,
700569,
703921,
707281,
710649,
714025,
717409,
720801,
724201,
727609,
731025,
734449,
737881,
741321,
744769,
748225,
751689,
755161,
758641,
762129,
765625,
769129,
772641,
776161,
779689,
783225,
786769,
790321,
793881,
797449,
801025,
804609,
808201,
811801,
815409,
819025,
822649,
826281,
829921,
833569,
837225,
840889,
844561,
848241,
851929,
855625,
859329,
863041,
866761,
870489,
874225,
877969,
881721,
885481,
889249,
893025,
896809,
900601,
904401,
908209,
912025,
915849,
919681,
923521,
927369,
931225,
935089,
938961,
942841,
946729,
950625,
954529,
958441,
962361,
966289,
970225,
974169,
978121,
982081,
986049,
990025,
994009,
998001,
1002001,
1006009,
1010025,
1014049,
1018081,
1022121,
1026169,
1030225,
1034289,
1038361,
1042441,
1046529,
1050625,
1054729,
1058841,
1062961,
1067089,
1071225,
1075369,
1079521,
1083681,
1087849,
1092025,
1096209,
1100401,
1104601,
1108809,
1113025,
1117249,
1121481,
1125721,
1129969,
1134225,
1138489,
1142761,
1147041,
1151329,
1155625,
1159929,
1164241,
1168561,
1172889,
1177225,
1181569,
1185921,
1190281,
1194649,
1199025,
1203409,
1207801,
1212201,
1216609,
1221025,
1225449,
1229881,
1234321,
1238769,
1243225,
1247689,
1252161,
1256641,
1261129,
1265625,
1270129,
1274641,
1279161,
1283689,
1288225,
1292769,
1297321,
1301881,
1306449,
1311025,
1315609,
1320201,
1324801,
1329409,
1334025,
1338649,
1343281,
1347921,
1352569,
1357225,
1361889,
1366561,
1371241,
1375929,
1380625,
1385329,
1390041,
1394761,
1399489,
1404225,
1408969,
1413721,
1418481,
1423249,
1428025,
1432809,
1437601.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 15811388 values, from 1 to 999999898700625).
n\r | 0 | 1 |
2 | 0 | 15811388 | 2 |
3 | 5270463 | 10540925 | 0 | 3 |
4 | 0 | 15811388 | 0 | 0 | 4 |
5 | 3162278 | 6324555 | 0 | 0 | 6324555 | 5 |
6 | 0 | 10540925 | 0 | 5270463 | 0 | 0 | 6 |
7 | 2258770 | 4517539 | 4517539 | 0 | 4517540 | 0 | 0 | 7 |
8 | 0 | 15811388 | 0 | 0 | 0 | 0 | 0 | 0 | 8 |
9 | 5270463 | 3513641 | 0 | 0 | 3513642 | 0 | 0 | 3513642 | 0 | 9 |
10 | 0 | 6324555 | 0 | 0 | 0 | 3162278 | 0 | 0 | 0 | 6324555 | 10 |
11 | 1437399 | 2874797 | 0 | 2874798 | 2874798 | 2874798 | 0 | 0 | 0 | 2874798 | 0 |
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.