A number of the form n(n+1). more
The first 600 pronic numbers :
2,
6,
12,
20,
30,
42,
56,
72,
90,
110,
132,
156,
182,
210,
240,
272,
306,
342,
380,
420,
462,
506,
552,
600,
650,
702,
756,
812,
870,
930,
992,
1056,
1122,
1190,
1260,
1332,
1406,
1482,
1560,
1640,
1722,
1806,
1892,
1980,
2070,
2162,
2256,
2352,
2450,
2550,
2652,
2756,
2862,
2970,
3080,
3192,
3306,
3422,
3540,
3660,
3782,
3906,
4032,
4160,
4290,
4422,
4556,
4692,
4830,
4970,
5112,
5256,
5402,
5550,
5700,
5852,
6006,
6162,
6320,
6480,
6642,
6806,
6972,
7140,
7310,
7482,
7656,
7832,
8010,
8190,
8372,
8556,
8742,
8930,
9120,
9312,
9506,
9702,
9900,
10100,
10302,
10506,
10712,
10920,
11130,
11342,
11556,
11772,
11990,
12210,
12432,
12656,
12882,
13110,
13340,
13572,
13806,
14042,
14280,
14520,
14762,
15006,
15252,
15500,
15750,
16002,
16256,
16512,
16770,
17030,
17292,
17556,
17822,
18090,
18360,
18632,
18906,
19182,
19460,
19740,
20022,
20306,
20592,
20880,
21170,
21462,
21756,
22052,
22350,
22650,
22952,
23256,
23562,
23870,
24180,
24492,
24806,
25122,
25440,
25760,
26082,
26406,
26732,
27060,
27390,
27722,
28056,
28392,
28730,
29070,
29412,
29756,
30102,
30450,
30800,
31152,
31506,
31862,
32220,
32580,
32942,
33306,
33672,
34040,
34410,
34782,
35156,
35532,
35910,
36290,
36672,
37056,
37442,
37830,
38220,
38612,
39006,
39402,
39800,
40200,
40602,
41006,
41412,
41820,
42230,
42642,
43056,
43472,
43890,
44310,
44732,
45156,
45582,
46010,
46440,
46872,
47306,
47742,
48180,
48620,
49062,
49506,
49952,
50400,
50850,
51302,
51756,
52212,
52670,
53130,
53592,
54056,
54522,
54990,
55460,
55932,
56406,
56882,
57360,
57840,
58322,
58806,
59292,
59780,
60270,
60762,
61256,
61752,
62250,
62750,
63252,
63756,
64262,
64770,
65280,
65792,
66306,
66822,
67340,
67860,
68382,
68906,
69432,
69960,
70490,
71022,
71556,
72092,
72630,
73170,
73712,
74256,
74802,
75350,
75900,
76452,
77006,
77562,
78120,
78680,
79242,
79806,
80372,
80940,
81510,
82082,
82656,
83232,
83810,
84390,
84972,
85556,
86142,
86730,
87320,
87912,
88506,
89102,
89700,
90300,
90902,
91506,
92112,
92720,
93330,
93942,
94556,
95172,
95790,
96410,
97032,
97656,
98282,
98910,
99540,
100172,
100806,
101442,
102080,
102720,
103362,
104006,
104652,
105300,
105950,
106602,
107256,
107912,
108570,
109230,
109892,
110556,
111222,
111890,
112560,
113232,
113906,
114582,
115260,
115940,
116622,
117306,
117992,
118680,
119370,
120062,
120756,
121452,
122150,
122850,
123552,
124256,
124962,
125670,
126380,
127092,
127806,
128522,
129240,
129960,
130682,
131406,
132132,
132860,
133590,
134322,
135056,
135792,
136530,
137270,
138012,
138756,
139502,
140250,
141000,
141752,
142506,
143262,
144020,
144780,
145542,
146306,
147072,
147840,
148610,
149382,
150156,
150932,
151710,
152490,
153272,
154056,
154842,
155630,
156420,
157212,
158006,
158802,
159600,
160400,
161202,
162006,
162812,
163620,
164430,
165242,
166056,
166872,
167690,
168510,
169332,
170156,
170982,
171810,
172640,
173472,
174306,
175142,
175980,
176820,
177662,
178506,
179352,
180200,
181050,
181902,
182756,
183612,
184470,
185330,
186192,
187056,
187922,
188790,
189660,
190532,
191406,
192282,
193160,
194040,
194922,
195806,
196692,
197580,
198470,
199362,
200256,
201152,
202050,
202950,
203852,
204756,
205662,
206570,
207480,
208392,
209306,
210222,
211140,
212060,
212982,
213906,
214832,
215760,
216690,
217622,
218556,
219492,
220430,
221370,
222312,
223256,
224202,
225150,
226100,
227052,
228006,
228962,
229920,
230880,
231842,
232806,
233772,
234740,
235710,
236682,
237656,
238632,
239610,
240590,
241572,
242556,
243542,
244530,
245520,
246512,
247506,
248502,
249500,
250500,
251502,
252506,
253512,
254520,
255530,
256542,
257556,
258572,
259590,
260610,
261632,
262656,
263682,
264710,
265740,
266772,
267806,
268842,
269880,
270920,
271962,
273006,
274052,
275100,
276150,
277202,
278256,
279312,
280370,
281430,
282492,
283556,
284622,
285690,
286760,
287832,
288906,
289982,
291060,
292140,
293222,
294306,
295392,
296480,
297570,
298662,
299756,
300852,
301950,
303050,
304152,
305256,
306362,
307470,
308580,
309692,
310806,
311922,
313040,
314160,
315282,
316406,
317532,
318660,
319790,
320922,
322056,
323192,
324330,
325470,
326612,
327756,
328902,
330050,
331200,
332352,
333506,
334662,
335820,
336980,
338142,
339306,
340472,
341640,
342810,
343982,
345156,
346332,
347510,
348690,
349872,
351056,
352242,
353430,
354620,
355812,
357006,
358202,
359400,
360600.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 31622776 values, from 2 to 999999993568952).
n\r | 0 | 1 |
2 | 31622776 | 0 | 2 |
3 | 21081850 | 0 | 10540926 | 3 |
4 | 15811388 | 0 | 15811388 | 0 | 4 |
5 | 12649110 | 6324555 | 12649111 | 0 | 0 | 5 |
6 | 21081850 | 0 | 10540926 | 0 | 0 | 0 | 6 |
7 | 9035078 | 0 | 9035079 | 0 | 0 | 4517540 | 9035079 | 7 |
8 | 7905694 | 0 | 7905694 | 0 | 7905694 | 0 | 7905694 | 0 | 8 |
9 | 7027282 | 0 | 10540926 | 7027284 | 0 | 0 | 7027284 | 0 | 0 | 9 |
10 | 12649110 | 0 | 12649111 | 0 | 0 | 0 | 6324555 | 0 | 0 | 0 | 10 |
11 | 5749594 | 5749596 | 5749596 | 0 | 0 | 0 | 5749596 | 0 | 2874798 | 5749596 | 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.