Figurate numbers of the form 5n(n-1)/2 + 1. more
The first 600 centered pentagonal numbers :
1,
6,
16,
31,
51,
76,
106,
141,
181,
226,
276,
331,
391,
456,
526,
601,
681,
766,
856,
951,
1051,
1156,
1266,
1381,
1501,
1626,
1756,
1891,
2031,
2176,
2326,
2481,
2641,
2806,
2976,
3151,
3331,
3516,
3706,
3901,
4101,
4306,
4516,
4731,
4951,
5176,
5406,
5641,
5881,
6126,
6376,
6631,
6891,
7156,
7426,
7701,
7981,
8266,
8556,
8851,
9151,
9456,
9766,
10081,
10401,
10726,
11056,
11391,
11731,
12076,
12426,
12781,
13141,
13506,
13876,
14251,
14631,
15016,
15406,
15801,
16201,
16606,
17016,
17431,
17851,
18276,
18706,
19141,
19581,
20026,
20476,
20931,
21391,
21856,
22326,
22801,
23281,
23766,
24256,
24751,
25251,
25756,
26266,
26781,
27301,
27826,
28356,
28891,
29431,
29976,
30526,
31081,
31641,
32206,
32776,
33351,
33931,
34516,
35106,
35701,
36301,
36906,
37516,
38131,
38751,
39376,
40006,
40641,
41281,
41926,
42576,
43231,
43891,
44556,
45226,
45901,
46581,
47266,
47956,
48651,
49351,
50056,
50766,
51481,
52201,
52926,
53656,
54391,
55131,
55876,
56626,
57381,
58141,
58906,
59676,
60451,
61231,
62016,
62806,
63601,
64401,
65206,
66016,
66831,
67651,
68476,
69306,
70141,
70981,
71826,
72676,
73531,
74391,
75256,
76126,
77001,
77881,
78766,
79656,
80551,
81451,
82356,
83266,
84181,
85101,
86026,
86956,
87891,
88831,
89776,
90726,
91681,
92641,
93606,
94576,
95551,
96531,
97516,
98506,
99501,
100501,
101506,
102516,
103531,
104551,
105576,
106606,
107641,
108681,
109726,
110776,
111831,
112891,
113956,
115026,
116101,
117181,
118266,
119356,
120451,
121551,
122656,
123766,
124881,
126001,
127126,
128256,
129391,
130531,
131676,
132826,
133981,
135141,
136306,
137476,
138651,
139831,
141016,
142206,
143401,
144601,
145806,
147016,
148231,
149451,
150676,
151906,
153141,
154381,
155626,
156876,
158131,
159391,
160656,
161926,
163201,
164481,
165766,
167056,
168351,
169651,
170956,
172266,
173581,
174901,
176226,
177556,
178891,
180231,
181576,
182926,
184281,
185641,
187006,
188376,
189751,
191131,
192516,
193906,
195301,
196701,
198106,
199516,
200931,
202351,
203776,
205206,
206641,
208081,
209526,
210976,
212431,
213891,
215356,
216826,
218301,
219781,
221266,
222756,
224251,
225751,
227256,
228766,
230281,
231801,
233326,
234856,
236391,
237931,
239476,
241026,
242581,
244141,
245706,
247276,
248851,
250431,
252016,
253606,
255201,
256801,
258406,
260016,
261631,
263251,
264876,
266506,
268141,
269781,
271426,
273076,
274731,
276391,
278056,
279726,
281401,
283081,
284766,
286456,
288151,
289851,
291556,
293266,
294981,
296701,
298426,
300156,
301891,
303631,
305376,
307126,
308881,
310641,
312406,
314176,
315951,
317731,
319516,
321306,
323101,
324901,
326706,
328516,
330331,
332151,
333976,
335806,
337641,
339481,
341326,
343176,
345031,
346891,
348756,
350626,
352501,
354381,
356266,
358156,
360051,
361951,
363856,
365766,
367681,
369601,
371526,
373456,
375391,
377331,
379276,
381226,
383181,
385141,
387106,
389076,
391051,
393031,
395016,
397006,
399001,
401001,
403006,
405016,
407031,
409051,
411076,
413106,
415141,
417181,
419226,
421276,
423331,
425391,
427456,
429526,
431601,
433681,
435766,
437856,
439951,
442051,
444156,
446266,
448381,
450501,
452626,
454756,
456891,
459031,
461176,
463326,
465481,
467641,
469806,
471976,
474151,
476331,
478516,
480706,
482901,
485101,
487306,
489516,
491731,
493951,
496176,
498406,
500641,
502881,
505126,
507376,
509631,
511891,
514156,
516426,
518701,
520981,
523266,
525556,
527851,
530151,
532456,
534766,
537081,
539401,
541726,
544056,
546391,
548731,
551076,
553426,
555781,
558141,
560506,
562876,
565251,
567631,
570016,
572406,
574801,
577201,
579606,
582016,
584431,
586851,
589276,
591706,
594141,
596581,
599026,
601476,
603931,
606391,
608856,
611326,
613801,
616281,
618766,
621256,
623751,
626251,
628756,
631266,
633781,
636301,
638826,
641356,
643891,
646431,
648976,
651526,
654081,
656641,
659206,
661776,
664351,
666931,
669516,
672106,
674701,
677301,
679906,
682516,
685131,
687751,
690376,
693006,
695641,
698281,
700926,
703576,
706231,
708891,
711556,
714226,
716901,
719581,
722266,
724956,
727651,
730351,
733056,
735766,
738481,
741201,
743926,
746656,
749391,
752131,
754876,
757626,
760381,
763141,
765906,
768676,
771451,
774231,
777016,
779806,
782601,
785401,
788206,
791016,
793831,
796651,
799476,
802306,
805141,
807981,
810826,
813676,
816531,
819391,
822256,
825126,
828001,
830881,
833766,
836656,
839551,
842451,
845356,
848266,
851181,
854101,
857026,
859956,
862891,
865831,
868776,
871726,
874681,
877641,
880606,
883576,
886551,
889531,
892516,
895506,
898501.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 20000000 values, from 1 to 999999950000001).
n\r | 0 | 1 |
2 | 10000000 | 10000000 | 2 |
3 | 6666667 | 13333333 | 0 | 3 |
4 | 5000000 | 5000000 | 5000000 | 5000000 | 4 |
5 | 0 | 20000000 | 0 | 0 | 0 | 5 |
6 | 3333333 | 6666666 | 0 | 3333334 | 6666667 | 0 | 6 |
7 | 0 | 5714285 | 5714286 | 2857143 | 0 | 0 | 5714286 | 7 |
8 | 2500000 | 2500000 | 2500000 | 2500000 | 2500000 | 2500000 | 2500000 | 2500000 | 8 |
9 | 0 | 4444445 | 0 | 0 | 4444444 | 0 | 6666667 | 4444444 | 0 | 9 |
10 | 0 | 10000000 | 0 | 0 | 0 | 0 | 10000000 | 0 | 0 | 0 | 10 |
11 | 0 | 3636363 | 0 | 0 | 0 | 3636364 | 3636363 | 3636364 | 0 | 3636364 | 1818182 |
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.