Figurate numbers of the form n(4n-3). more
The first 600 decagonal numbers :
1,
10,
27,
52,
85,
126,
175,
232,
297,
370,
451,
540,
637,
742,
855,
976,
1105,
1242,
1387,
1540,
1701,
1870,
2047,
2232,
2425,
2626,
2835,
3052,
3277,
3510,
3751,
4000,
4257,
4522,
4795,
5076,
5365,
5662,
5967,
6280,
6601,
6930,
7267,
7612,
7965,
8326,
8695,
9072,
9457,
9850,
10251,
10660,
11077,
11502,
11935,
12376,
12825,
13282,
13747,
14220,
14701,
15190,
15687,
16192,
16705,
17226,
17755,
18292,
18837,
19390,
19951,
20520,
21097,
21682,
22275,
22876,
23485,
24102,
24727,
25360,
26001,
26650,
27307,
27972,
28645,
29326,
30015,
30712,
31417,
32130,
32851,
33580,
34317,
35062,
35815,
36576,
37345,
38122,
38907,
39700,
40501,
41310,
42127,
42952,
43785,
44626,
45475,
46332,
47197,
48070,
48951,
49840,
50737,
51642,
52555,
53476,
54405,
55342,
56287,
57240,
58201,
59170,
60147,
61132,
62125,
63126,
64135,
65152,
66177,
67210,
68251,
69300,
70357,
71422,
72495,
73576,
74665,
75762,
76867,
77980,
79101,
80230,
81367,
82512,
83665,
84826,
85995,
87172,
88357,
89550,
90751,
91960,
93177,
94402,
95635,
96876,
98125,
99382,
100647,
101920,
103201,
104490,
105787,
107092,
108405,
109726,
111055,
112392,
113737,
115090,
116451,
117820,
119197,
120582,
121975,
123376,
124785,
126202,
127627,
129060,
130501,
131950,
133407,
134872,
136345,
137826,
139315,
140812,
142317,
143830,
145351,
146880,
148417,
149962,
151515,
153076,
154645,
156222,
157807,
159400,
161001,
162610,
164227,
165852,
167485,
169126,
170775,
172432,
174097,
175770,
177451,
179140,
180837,
182542,
184255,
185976,
187705,
189442,
191187,
192940,
194701,
196470,
198247,
200032,
201825,
203626,
205435,
207252,
209077,
210910,
212751,
214600,
216457,
218322,
220195,
222076,
223965,
225862,
227767,
229680,
231601,
233530,
235467,
237412,
239365,
241326,
243295,
245272,
247257,
249250,
251251,
253260,
255277,
257302,
259335,
261376,
263425,
265482,
267547,
269620,
271701,
273790,
275887,
277992,
280105,
282226,
284355,
286492,
288637,
290790,
292951,
295120,
297297,
299482,
301675,
303876,
306085,
308302,
310527,
312760,
315001,
317250,
319507,
321772,
324045,
326326,
328615,
330912,
333217,
335530,
337851,
340180,
342517,
344862,
347215,
349576,
351945,
354322,
356707,
359100,
361501,
363910,
366327,
368752,
371185,
373626,
376075,
378532,
380997,
383470,
385951,
388440,
390937,
393442,
395955,
398476,
401005,
403542,
406087,
408640,
411201,
413770,
416347,
418932,
421525,
424126,
426735,
429352,
431977,
434610,
437251,
439900,
442557,
445222,
447895,
450576,
453265,
455962,
458667,
461380,
464101,
466830,
469567,
472312,
475065,
477826,
480595,
483372,
486157,
488950,
491751,
494560,
497377,
500202,
503035,
505876,
508725,
511582,
514447,
517320,
520201,
523090,
525987,
528892,
531805,
534726,
537655,
540592,
543537,
546490,
549451,
552420,
555397,
558382,
561375,
564376,
567385,
570402,
573427,
576460,
579501,
582550,
585607,
588672,
591745,
594826,
597915,
601012,
604117,
607230,
610351,
613480,
616617,
619762,
622915,
626076,
629245,
632422,
635607,
638800,
642001,
645210,
648427,
651652,
654885,
658126,
661375,
664632,
667897,
671170,
674451,
677740,
681037,
684342,
687655,
690976,
694305,
697642,
700987,
704340,
707701,
711070,
714447,
717832,
721225,
724626,
728035,
731452,
734877,
738310,
741751,
745200,
748657,
752122,
755595,
759076,
762565,
766062,
769567,
773080,
776601,
780130,
783667,
787212,
790765,
794326,
797895,
801472,
805057,
808650,
812251,
815860,
819477,
823102,
826735,
830376,
834025,
837682,
841347,
845020,
848701,
852390,
856087,
859792,
863505,
867226,
870955,
874692,
878437,
882190,
885951,
889720,
893497,
897282,
901075,
904876,
908685,
912502,
916327,
920160,
924001,
927850,
931707,
935572,
939445,
943326,
947215,
951112,
955017,
958930,
962851,
966780,
970717,
974662,
978615,
982576,
986545,
990522,
994507,
998500,
1002501,
1006510,
1010527,
1014552,
1018585,
1022626,
1026675,
1030732,
1034797,
1038870,
1042951,
1047040,
1051137,
1055242,
1059355,
1063476,
1067605,
1071742,
1075887,
1080040,
1084201,
1088370,
1092547,
1096732,
1100925,
1105126,
1109335,
1113552,
1117777,
1122010,
1126251,
1130500,
1134757,
1139022,
1143295,
1147576,
1151865,
1156162,
1160467,
1164780,
1169101,
1173430,
1177767,
1182112,
1186465,
1190826,
1195195,
1199572,
1203957,
1208350,
1212751,
1217160,
1221577,
1226002,
1230435,
1234876,
1239325,
1243782,
1248247,
1252720,
1257201,
1261690,
1266187,
1270692,
1275205,
1279726,
1284255,
1288792,
1293337,
1297890,
1302451,
1307020,
1311597,
1316182,
1320775,
1325376,
1329985,
1334602,
1339227,
1343860,
1348501,
1353150,
1357807,
1362472,
1367145,
1371826,
1376515,
1381212,
1385917,
1390630,
1395351,
1400080,
1404817,
1409562,
1414315,
1419076,
1423845,
1428622,
1433407,
1438200.
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 999999914512012).
n\r | 0 | 1 |
2 | 7905694 | 7905694 | 2 |
3 | 5270462 | 10540926 | 0 | 3 |
4 | 3952847 | 3952847 | 3952847 | 3952847 | 4 |
5 | 6324555 | 3162278 | 6324555 | 0 | 0 | 5 |
6 | 2635231 | 5270463 | 0 | 2635231 | 5270463 | 0 | 6 |
7 | 4517538 | 4517540 | 0 | 4517540 | 0 | 0 | 2258770 | 7 |
8 | 1976423 | 1976424 | 1976424 | 1976424 | 1976424 | 1976423 | 1976423 | 1976423 | 8 |
9 | 5270462 | 3513642 | 0 | 0 | 3513642 | 0 | 0 | 3513642 | 0 | 9 |
10 | 3162277 | 1581139 | 3162278 | 0 | 0 | 3162278 | 1581139 | 3162277 | 0 | 0 | 10 |
11 | 2874797 | 2874798 | 0 | 0 | 0 | 2874798 | 0 | 1437399 | 2874798 | 0 | 2874798 |
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.