Figurate numbers of the form 9n(n-1)/2 + 1. more
The first 600 centered nonagonal numbers :
1,
10,
28,
55,
91,
136,
190,
253,
325,
406,
496,
595,
703,
820,
946,
1081,
1225,
1378,
1540,
1711,
1891,
2080,
2278,
2485,
2701,
2926,
3160,
3403,
3655,
3916,
4186,
4465,
4753,
5050,
5356,
5671,
5995,
6328,
6670,
7021,
7381,
7750,
8128,
8515,
8911,
9316,
9730,
10153,
10585,
11026,
11476,
11935,
12403,
12880,
13366,
13861,
14365,
14878,
15400,
15931,
16471,
17020,
17578,
18145,
18721,
19306,
19900,
20503,
21115,
21736,
22366,
23005,
23653,
24310,
24976,
25651,
26335,
27028,
27730,
28441,
29161,
29890,
30628,
31375,
32131,
32896,
33670,
34453,
35245,
36046,
36856,
37675,
38503,
39340,
40186,
41041,
41905,
42778,
43660,
44551,
45451,
46360,
47278,
48205,
49141,
50086,
51040,
52003,
52975,
53956,
54946,
55945,
56953,
57970,
58996,
60031,
61075,
62128,
63190,
64261,
65341,
66430,
67528,
68635,
69751,
70876,
72010,
73153,
74305,
75466,
76636,
77815,
79003,
80200,
81406,
82621,
83845,
85078,
86320,
87571,
88831,
90100,
91378,
92665,
93961,
95266,
96580,
97903,
99235,
100576,
101926,
103285,
104653,
106030,
107416,
108811,
110215,
111628,
113050,
114481,
115921,
117370,
118828,
120295,
121771,
123256,
124750,
126253,
127765,
129286,
130816,
132355,
133903,
135460,
137026,
138601,
140185,
141778,
143380,
144991,
146611,
148240,
149878,
151525,
153181,
154846,
156520,
158203,
159895,
161596,
163306,
165025,
166753,
168490,
170236,
171991,
173755,
175528,
177310,
179101,
180901,
182710,
184528,
186355,
188191,
190036,
191890,
193753,
195625,
197506,
199396,
201295,
203203,
205120,
207046,
208981,
210925,
212878,
214840,
216811,
218791,
220780,
222778,
224785,
226801,
228826,
230860,
232903,
234955,
237016,
239086,
241165,
243253,
245350,
247456,
249571,
251695,
253828,
255970,
258121,
260281,
262450,
264628,
266815,
269011,
271216,
273430,
275653,
277885,
280126,
282376,
284635,
286903,
289180,
291466,
293761,
296065,
298378,
300700,
303031,
305371,
307720,
310078,
312445,
314821,
317206,
319600,
322003,
324415,
326836,
329266,
331705,
334153,
336610,
339076,
341551,
344035,
346528,
349030,
351541,
354061,
356590,
359128,
361675,
364231,
366796,
369370,
371953,
374545,
377146,
379756,
382375,
385003,
387640,
390286,
392941,
395605,
398278,
400960,
403651,
406351,
409060,
411778,
414505,
417241,
419986,
422740,
425503,
428275,
431056,
433846,
436645,
439453,
442270,
445096,
447931,
450775,
453628,
456490,
459361,
462241,
465130,
468028,
470935,
473851,
476776,
479710,
482653,
485605,
488566,
491536,
494515,
497503,
500500,
503506,
506521,
509545,
512578,
515620,
518671,
521731,
524800,
527878,
530965,
534061,
537166,
540280,
543403,
546535,
549676,
552826,
555985,
559153,
562330,
565516,
568711,
571915,
575128,
578350,
581581,
584821,
588070,
591328,
594595,
597871,
601156,
604450,
607753,
611065,
614386,
617716,
621055,
624403,
627760,
631126,
634501,
637885,
641278,
644680,
648091,
651511,
654940,
658378,
661825,
665281,
668746,
672220,
675703,
679195,
682696,
686206,
689725,
693253,
696790,
700336,
703891,
707455,
711028,
714610,
718201,
721801,
725410,
729028,
732655,
736291,
739936,
743590,
747253,
750925,
754606,
758296,
761995,
765703,
769420,
773146,
776881,
780625,
784378,
788140,
791911,
795691,
799480,
803278,
807085,
810901,
814726,
818560,
822403,
826255,
830116,
833986,
837865,
841753,
845650,
849556,
853471,
857395,
861328,
865270,
869221,
873181,
877150,
881128,
885115,
889111,
893116,
897130,
901153,
905185,
909226,
913276,
917335,
921403,
925480,
929566,
933661,
937765,
941878,
946000,
950131,
954271,
958420,
962578,
966745,
970921,
975106,
979300,
983503,
987715,
991936,
996166,
1000405,
1004653,
1008910,
1013176,
1017451,
1021735,
1026028,
1030330,
1034641,
1038961,
1043290,
1047628,
1051975,
1056331,
1060696,
1065070,
1069453,
1073845,
1078246,
1082656,
1087075,
1091503,
1095940,
1100386,
1104841,
1109305,
1113778,
1118260,
1122751,
1127251,
1131760,
1136278,
1140805,
1145341,
1149886,
1154440,
1159003,
1163575,
1168156,
1172746,
1177345,
1181953,
1186570,
1191196,
1195831,
1200475,
1205128,
1209790,
1214461,
1219141,
1223830,
1228528,
1233235,
1237951,
1242676,
1247410,
1252153,
1256905,
1261666,
1266436,
1271215,
1276003,
1280800,
1285606,
1290421,
1295245,
1300078,
1304920,
1309771,
1314631,
1319500,
1324378,
1329265,
1334161,
1339066,
1343980,
1348903,
1353835,
1358776,
1363726,
1368685,
1373653,
1378630,
1383616,
1388611,
1393615,
1398628,
1403650,
1408681,
1413721,
1418770,
1423828,
1428895,
1433971,
1439056,
1444150,
1449253,
1454365,
1459486,
1464616,
1469755,
1474903,
1480060,
1485226,
1490401,
1495585,
1500778,
1505980,
1511191,
1516411,
1521640,
1526878,
1532125,
1537381,
1542646,
1547920,
1553203,
1558495,
1563796,
1569106,
1574425,
1579753,
1585090,
1590436,
1595791,
1601155,
1606528,
1611910,
1617301.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 14907120 values, from 1 to 999999953042761).
n\r | 0 | 1 |
2 | 7453560 | 7453560 | 2 |
3 | 0 | 14907120 | 0 | 3 |
4 | 3726780 | 3726780 | 3726780 | 3726780 | 4 |
5 | 5962848 | 5962848 | 0 | 2981424 | 0 | 5 |
6 | 0 | 7453560 | 0 | 0 | 7453560 | 0 | 6 |
7 | 4259177 | 4259177 | 0 | 4259177 | 0 | 0 | 2129589 | 7 |
8 | 1863390 | 1863390 | 1863390 | 1863390 | 1863390 | 1863390 | 1863390 | 1863390 | 8 |
9 | 0 | 14907120 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 9 |
10 | 2981424 | 2981424 | 0 | 1490712 | 0 | 2981424 | 2981424 | 0 | 1490712 | 0 | 10 |
11 | 2710386 | 2710385 | 0 | 2710386 | 1355193 | 0 | 2710385 | 0 | 0 | 0 | 2710385 |
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.