A figurate number of form n(3n-1)/2. more
The first 600 pentagonal numbers :
1,
5,
12,
22,
35,
51,
70,
92,
117,
145,
176,
210,
247,
287,
330,
376,
425,
477,
532,
590,
651,
715,
782,
852,
925,
1001,
1080,
1162,
1247,
1335,
1426,
1520,
1617,
1717,
1820,
1926,
2035,
2147,
2262,
2380,
2501,
2625,
2752,
2882,
3015,
3151,
3290,
3432,
3577,
3725,
3876,
4030,
4187,
4347,
4510,
4676,
4845,
5017,
5192,
5370,
5551,
5735,
5922,
6112,
6305,
6501,
6700,
6902,
7107,
7315,
7526,
7740,
7957,
8177,
8400,
8626,
8855,
9087,
9322,
9560,
9801,
10045,
10292,
10542,
10795,
11051,
11310,
11572,
11837,
12105,
12376,
12650,
12927,
13207,
13490,
13776,
14065,
14357,
14652,
14950,
15251,
15555,
15862,
16172,
16485,
16801,
17120,
17442,
17767,
18095,
18426,
18760,
19097,
19437,
19780,
20126,
20475,
20827,
21182,
21540,
21901,
22265,
22632,
23002,
23375,
23751,
24130,
24512,
24897,
25285,
25676,
26070,
26467,
26867,
27270,
27676,
28085,
28497,
28912,
29330,
29751,
30175,
30602,
31032,
31465,
31901,
32340,
32782,
33227,
33675,
34126,
34580,
35037,
35497,
35960,
36426,
36895,
37367,
37842,
38320,
38801,
39285,
39772,
40262,
40755,
41251,
41750,
42252,
42757,
43265,
43776,
44290,
44807,
45327,
45850,
46376,
46905,
47437,
47972,
48510,
49051,
49595,
50142,
50692,
51245,
51801,
52360,
52922,
53487,
54055,
54626,
55200,
55777,
56357,
56940,
57526,
58115,
58707,
59302,
59900,
60501,
61105,
61712,
62322,
62935,
63551,
64170,
64792,
65417,
66045,
66676,
67310,
67947,
68587,
69230,
69876,
70525,
71177,
71832,
72490,
73151,
73815,
74482,
75152,
75825,
76501,
77180,
77862,
78547,
79235,
79926,
80620,
81317,
82017,
82720,
83426,
84135,
84847,
85562,
86280,
87001,
87725,
88452,
89182,
89915,
90651,
91390,
92132,
92877,
93625,
94376,
95130,
95887,
96647,
97410,
98176,
98945,
99717,
100492,
101270,
102051,
102835,
103622,
104412,
105205,
106001,
106800,
107602,
108407,
109215,
110026,
110840,
111657,
112477,
113300,
114126,
114955,
115787,
116622,
117460,
118301,
119145,
119992,
120842,
121695,
122551,
123410,
124272,
125137,
126005,
126876,
127750,
128627,
129507,
130390,
131276,
132165,
133057,
133952,
134850,
135751,
136655,
137562,
138472,
139385,
140301,
141220,
142142,
143067,
143995,
144926,
145860,
146797,
147737,
148680,
149626,
150575,
151527,
152482,
153440,
154401,
155365,
156332,
157302,
158275,
159251,
160230,
161212,
162197,
163185,
164176,
165170,
166167,
167167,
168170,
169176,
170185,
171197,
172212,
173230,
174251,
175275,
176302,
177332,
178365,
179401,
180440,
181482,
182527,
183575,
184626,
185680,
186737,
187797,
188860,
189926,
190995,
192067,
193142,
194220,
195301,
196385,
197472,
198562,
199655,
200751,
201850,
202952,
204057,
205165,
206276,
207390,
208507,
209627,
210750,
211876,
213005,
214137,
215272,
216410,
217551,
218695,
219842,
220992,
222145,
223301,
224460,
225622,
226787,
227955,
229126,
230300,
231477,
232657,
233840,
235026,
236215,
237407,
238602,
239800,
241001,
242205,
243412,
244622,
245835,
247051,
248270,
249492,
250717,
251945,
253176,
254410,
255647,
256887,
258130,
259376,
260625,
261877,
263132,
264390,
265651,
266915,
268182,
269452,
270725,
272001,
273280,
274562,
275847,
277135,
278426,
279720,
281017,
282317,
283620,
284926,
286235,
287547,
288862,
290180,
291501,
292825,
294152,
295482,
296815,
298151,
299490,
300832,
302177,
303525,
304876,
306230,
307587,
308947,
310310,
311676,
313045,
314417,
315792,
317170,
318551,
319935,
321322,
322712,
324105,
325501,
326900,
328302,
329707,
331115,
332526,
333940,
335357,
336777,
338200,
339626,
341055,
342487,
343922,
345360,
346801,
348245,
349692,
351142,
352595,
354051,
355510,
356972,
358437,
359905,
361376,
362850,
364327,
365807,
367290,
368776,
370265,
371757,
373252,
374750,
376251,
377755,
379262,
380772,
382285,
383801,
385320,
386842,
388367,
389895,
391426,
392960,
394497,
396037,
397580,
399126,
400675,
402227,
403782,
405340,
406901,
408465,
410032,
411602,
413175,
414751,
416330,
417912,
419497,
421085,
422676,
424270,
425867,
427467,
429070,
430676,
432285,
433897,
435512,
437130,
438751,
440375,
442002,
443632,
445265,
446901,
448540,
450182,
451827,
453475,
455126,
456780,
458437,
460097,
461760,
463426,
465095,
466767,
468442,
470120,
471801,
473485,
475172,
476862,
478555,
480251,
481950,
483652,
485357,
487065,
488776,
490490,
492207,
493927,
495650,
497376,
499105,
500837,
502572,
504310,
506051,
507795,
509542,
511292,
513045,
514801,
516560,
518322,
520087,
521855,
523626,
525400,
527177,
528957,
530740,
532526,
534315,
536107,
537902,
539700.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 25819889 values, from 1 to 999999989048537).
n\r | 0 | 1 |
2 | 12909944 | 12909945 | 2 |
3 | 8606629 | 8606630 | 8606630 | 3 |
4 | 6454972 | 6454973 | 6454972 | 6454972 | 4 |
5 | 10327955 | 5163978 | 10327956 | 0 | 0 | 5 |
6 | 4303315 | 4303315 | 4303314 | 4303314 | 4303315 | 4303316 | 6 |
7 | 7377110 | 7377112 | 3688555 | 0 | 0 | 7377112 | 0 | 7 |
8 | 3227486 | 3227487 | 3227486 | 3227486 | 3227486 | 3227486 | 3227486 | 3227486 | 8 |
9 | 2868876 | 2868877 | 2868876 | 2868877 | 2868877 | 2868877 | 2868876 | 2868876 | 2868877 | 9 |
10 | 5163977 | 2581990 | 5163979 | 0 | 0 | 5163978 | 2581988 | 5163977 | 0 | 0 | 10 |
11 | 4694525 | 4694526 | 4694525 | 0 | 4694525 | 2347263 | 0 | 4694525 | 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.