Figurate numbers of the form 6n(n-1) + 1. more
The first 600 star numbers :
1,
13,
37,
73,
121,
181,
253,
337,
433,
541,
661,
793,
937,
1093,
1261,
1441,
1633,
1837,
2053,
2281,
2521,
2773,
3037,
3313,
3601,
3901,
4213,
4537,
4873,
5221,
5581,
5953,
6337,
6733,
7141,
7561,
7993,
8437,
8893,
9361,
9841,
10333,
10837,
11353,
11881,
12421,
12973,
13537,
14113,
14701,
15301,
15913,
16537,
17173,
17821,
18481,
19153,
19837,
20533,
21241,
21961,
22693,
23437,
24193,
24961,
25741,
26533,
27337,
28153,
28981,
29821,
30673,
31537,
32413,
33301,
34201,
35113,
36037,
36973,
37921,
38881,
39853,
40837,
41833,
42841,
43861,
44893,
45937,
46993,
48061,
49141,
50233,
51337,
52453,
53581,
54721,
55873,
57037,
58213,
59401,
60601,
61813,
63037,
64273,
65521,
66781,
68053,
69337,
70633,
71941,
73261,
74593,
75937,
77293,
78661,
80041,
81433,
82837,
84253,
85681,
87121,
88573,
90037,
91513,
93001,
94501,
96013,
97537,
99073,
100621,
102181,
103753,
105337,
106933,
108541,
110161,
111793,
113437,
115093,
116761,
118441,
120133,
121837,
123553,
125281,
127021,
128773,
130537,
132313,
134101,
135901,
137713,
139537,
141373,
143221,
145081,
146953,
148837,
150733,
152641,
154561,
156493,
158437,
160393,
162361,
164341,
166333,
168337,
170353,
172381,
174421,
176473,
178537,
180613,
182701,
184801,
186913,
189037,
191173,
193321,
195481,
197653,
199837,
202033,
204241,
206461,
208693,
210937,
213193,
215461,
217741,
220033,
222337,
224653,
226981,
229321,
231673,
234037,
236413,
238801,
241201,
243613,
246037,
248473,
250921,
253381,
255853,
258337,
260833,
263341,
265861,
268393,
270937,
273493,
276061,
278641,
281233,
283837,
286453,
289081,
291721,
294373,
297037,
299713,
302401,
305101,
307813,
310537,
313273,
316021,
318781,
321553,
324337,
327133,
329941,
332761,
335593,
338437,
341293,
344161,
347041,
349933,
352837,
355753,
358681,
361621,
364573,
367537,
370513,
373501,
376501,
379513,
382537,
385573,
388621,
391681,
394753,
397837,
400933,
404041,
407161,
410293,
413437,
416593,
419761,
422941,
426133,
429337,
432553,
435781,
439021,
442273,
445537,
448813,
452101,
455401,
458713,
462037,
465373,
468721,
472081,
475453,
478837,
482233,
485641,
489061,
492493,
495937,
499393,
502861,
506341,
509833,
513337,
516853,
520381,
523921,
527473,
531037,
534613,
538201,
541801,
545413,
549037,
552673,
556321,
559981,
563653,
567337,
571033,
574741,
578461,
582193,
585937,
589693,
593461,
597241,
601033,
604837,
608653,
612481,
616321,
620173,
624037,
627913,
631801,
635701,
639613,
643537,
647473,
651421,
655381,
659353,
663337,
667333,
671341,
675361,
679393,
683437,
687493,
691561,
695641,
699733,
703837,
707953,
712081,
716221,
720373,
724537,
728713,
732901,
737101,
741313,
745537,
749773,
754021,
758281,
762553,
766837,
771133,
775441,
779761,
784093,
788437,
792793,
797161,
801541,
805933,
810337,
814753,
819181,
823621,
828073,
832537,
837013,
841501,
846001,
850513,
855037,
859573,
864121,
868681,
873253,
877837,
882433,
887041,
891661,
896293,
900937,
905593,
910261,
914941,
919633,
924337,
929053,
933781,
938521,
943273,
948037,
952813,
957601,
962401,
967213,
972037,
976873,
981721,
986581,
991453,
996337,
1001233,
1006141,
1011061,
1015993,
1020937,
1025893,
1030861,
1035841,
1040833,
1045837,
1050853,
1055881,
1060921,
1065973,
1071037,
1076113,
1081201,
1086301,
1091413,
1096537,
1101673,
1106821,
1111981,
1117153,
1122337,
1127533,
1132741,
1137961,
1143193,
1148437,
1153693,
1158961,
1164241,
1169533,
1174837,
1180153,
1185481,
1190821,
1196173,
1201537,
1206913,
1212301,
1217701,
1223113,
1228537,
1233973,
1239421,
1244881,
1250353,
1255837,
1261333,
1266841,
1272361,
1277893,
1283437,
1288993,
1294561,
1300141,
1305733,
1311337,
1316953,
1322581,
1328221,
1333873,
1339537,
1345213,
1350901,
1356601,
1362313,
1368037,
1373773,
1379521,
1385281,
1391053,
1396837,
1402633,
1408441,
1414261,
1420093,
1425937,
1431793,
1437661,
1443541,
1449433,
1455337,
1461253,
1467181,
1473121,
1479073,
1485037,
1491013,
1497001,
1503001,
1509013,
1515037,
1521073,
1527121,
1533181,
1539253,
1545337,
1551433,
1557541,
1563661,
1569793,
1575937,
1582093,
1588261,
1594441,
1600633,
1606837,
1613053,
1619281,
1625521,
1631773,
1638037,
1644313,
1650601,
1656901,
1663213,
1669537,
1675873,
1682221,
1688581,
1694953,
1701337,
1707733,
1714141,
1720561,
1726993,
1733437,
1739893,
1746361,
1752841,
1759333,
1765837,
1772353,
1778881,
1785421,
1791973,
1798537,
1805113,
1811701,
1818301,
1824913,
1831537,
1838173,
1844821,
1851481,
1858153,
1864837,
1871533,
1878241,
1884961,
1891693,
1898437,
1905193,
1911961,
1918741,
1925533,
1932337,
1939153,
1945981,
1952821,
1959673,
1966537,
1973413,
1980301,
1987201,
1994113,
2001037,
2007973,
2014921,
2021881,
2028853,
2035837,
2042833,
2049841,
2056861,
2063893,
2070937,
2077993,
2085061,
2092141,
2099233,
2106337,
2113453,
2120581,
2127721,
2134873,
2142037,
2149213,
2156401.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 12909944 values, from 1 to 999999847039153).
n\r | 0 | 1 |
2 | 0 | 12909944 | 2 |
3 | 0 | 12909944 | 0 | 3 |
4 | 0 | 12909944 | 0 | 0 | 4 |
5 | 0 | 5163977 | 2581989 | 5163978 | 0 | 5 |
6 | 0 | 12909944 | 0 | 0 | 0 | 0 | 6 |
7 | 0 | 3688555 | 3688556 | 1844278 | 0 | 0 | 3688555 | 7 |
8 | 0 | 6454972 | 0 | 0 | 0 | 6454972 | 0 | 0 | 8 |
9 | 0 | 8606629 | 0 | 0 | 4303315 | 0 | 0 | 0 | 0 | 9 |
10 | 0 | 5163977 | 0 | 5163978 | 0 | 0 | 0 | 2581989 | 0 | 0 | 10 |
11 | 2347262 | 2347263 | 2347263 | 0 | 2347263 | 1173631 | 0 | 2347262 | 0 | 0 | 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.