Figurate numbers of the form n(n+1)/2. more
The first 600 triangular numbers :
1,
3,
6,
10,
15,
21,
28,
36,
45,
55,
66,
78,
91,
105,
120,
136,
153,
171,
190,
210,
231,
253,
276,
300,
325,
351,
378,
406,
435,
465,
496,
528,
561,
595,
630,
666,
703,
741,
780,
820,
861,
903,
946,
990,
1035,
1081,
1128,
1176,
1225,
1275,
1326,
1378,
1431,
1485,
1540,
1596,
1653,
1711,
1770,
1830,
1891,
1953,
2016,
2080,
2145,
2211,
2278,
2346,
2415,
2485,
2556,
2628,
2701,
2775,
2850,
2926,
3003,
3081,
3160,
3240,
3321,
3403,
3486,
3570,
3655,
3741,
3828,
3916,
4005,
4095,
4186,
4278,
4371,
4465,
4560,
4656,
4753,
4851,
4950,
5050,
5151,
5253,
5356,
5460,
5565,
5671,
5778,
5886,
5995,
6105,
6216,
6328,
6441,
6555,
6670,
6786,
6903,
7021,
7140,
7260,
7381,
7503,
7626,
7750,
7875,
8001,
8128,
8256,
8385,
8515,
8646,
8778,
8911,
9045,
9180,
9316,
9453,
9591,
9730,
9870,
10011,
10153,
10296,
10440,
10585,
10731,
10878,
11026,
11175,
11325,
11476,
11628,
11781,
11935,
12090,
12246,
12403,
12561,
12720,
12880,
13041,
13203,
13366,
13530,
13695,
13861,
14028,
14196,
14365,
14535,
14706,
14878,
15051,
15225,
15400,
15576,
15753,
15931,
16110,
16290,
16471,
16653,
16836,
17020,
17205,
17391,
17578,
17766,
17955,
18145,
18336,
18528,
18721,
18915,
19110,
19306,
19503,
19701,
19900,
20100,
20301,
20503,
20706,
20910,
21115,
21321,
21528,
21736,
21945,
22155,
22366,
22578,
22791,
23005,
23220,
23436,
23653,
23871,
24090,
24310,
24531,
24753,
24976,
25200,
25425,
25651,
25878,
26106,
26335,
26565,
26796,
27028,
27261,
27495,
27730,
27966,
28203,
28441,
28680,
28920,
29161,
29403,
29646,
29890,
30135,
30381,
30628,
30876,
31125,
31375,
31626,
31878,
32131,
32385,
32640,
32896,
33153,
33411,
33670,
33930,
34191,
34453,
34716,
34980,
35245,
35511,
35778,
36046,
36315,
36585,
36856,
37128,
37401,
37675,
37950,
38226,
38503,
38781,
39060,
39340,
39621,
39903,
40186,
40470,
40755,
41041,
41328,
41616,
41905,
42195,
42486,
42778,
43071,
43365,
43660,
43956,
44253,
44551,
44850,
45150,
45451,
45753,
46056,
46360,
46665,
46971,
47278,
47586,
47895,
48205,
48516,
48828,
49141,
49455,
49770,
50086,
50403,
50721,
51040,
51360,
51681,
52003,
52326,
52650,
52975,
53301,
53628,
53956,
54285,
54615,
54946,
55278,
55611,
55945,
56280,
56616,
56953,
57291,
57630,
57970,
58311,
58653,
58996,
59340,
59685,
60031,
60378,
60726,
61075,
61425,
61776,
62128,
62481,
62835,
63190,
63546,
63903,
64261,
64620,
64980,
65341,
65703,
66066,
66430,
66795,
67161,
67528,
67896,
68265,
68635,
69006,
69378,
69751,
70125,
70500,
70876,
71253,
71631,
72010,
72390,
72771,
73153,
73536,
73920,
74305,
74691,
75078,
75466,
75855,
76245,
76636,
77028,
77421,
77815,
78210,
78606,
79003,
79401,
79800,
80200,
80601,
81003,
81406,
81810,
82215,
82621,
83028,
83436,
83845,
84255,
84666,
85078,
85491,
85905,
86320,
86736,
87153,
87571,
87990,
88410,
88831,
89253,
89676,
90100,
90525,
90951,
91378,
91806,
92235,
92665,
93096,
93528,
93961,
94395,
94830,
95266,
95703,
96141,
96580,
97020,
97461,
97903,
98346,
98790,
99235,
99681,
100128,
100576,
101025,
101475,
101926,
102378,
102831,
103285,
103740,
104196,
104653,
105111,
105570,
106030,
106491,
106953,
107416,
107880,
108345,
108811,
109278,
109746,
110215,
110685,
111156,
111628,
112101,
112575,
113050,
113526,
114003,
114481,
114960,
115440,
115921,
116403,
116886,
117370,
117855,
118341,
118828,
119316,
119805,
120295,
120786,
121278,
121771,
122265,
122760,
123256,
123753,
124251,
124750,
125250,
125751,
126253,
126756,
127260,
127765,
128271,
128778,
129286,
129795,
130305,
130816,
131328,
131841,
132355,
132870,
133386,
133903,
134421,
134940,
135460,
135981,
136503,
137026,
137550,
138075,
138601,
139128,
139656,
140185,
140715,
141246,
141778,
142311,
142845,
143380,
143916,
144453,
144991,
145530,
146070,
146611,
147153,
147696,
148240,
148785,
149331,
149878,
150426,
150975,
151525,
152076,
152628,
153181,
153735,
154290,
154846,
155403,
155961,
156520,
157080,
157641,
158203,
158766,
159330,
159895,
160461,
161028,
161596,
162165,
162735,
163306,
163878,
164451,
165025,
165600,
166176,
166753,
167331,
167910,
168490,
169071,
169653,
170236,
170820,
171405,
171991,
172578,
173166,
173755,
174345,
174936,
175528,
176121,
176715,
177310,
177906,
178503,
179101,
179700,
180300.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 44721359 values, from 1 to 999999997764120).
n\r | 0 | 1 |
2 | 22360679 | 22360680 | 2 |
3 | 29814239 | 14907120 | 0 | 3 |
4 | 11180339 | 11180340 | 11180340 | 11180340 | 4 |
5 | 17888543 | 17888544 | 0 | 8944272 | 0 | 5 |
6 | 14907119 | 7453560 | 0 | 14907120 | 7453560 | 0 | 6 |
7 | 12777530 | 12777531 | 0 | 12777532 | 0 | 0 | 6388766 | 7 |
8 | 5590169 | 5590170 | 5590170 | 5590170 | 5590170 | 5590170 | 5590170 | 5590170 | 8 |
9 | 9938079 | 14907120 | 0 | 9938080 | 0 | 0 | 9938080 | 0 | 0 | 9 |
10 | 8944271 | 8944272 | 0 | 4472136 | 0 | 8944272 | 8944272 | 0 | 4472136 | 0 | 10 |
11 | 8131156 | 8131157 | 0 | 8131156 | 4065578 | 0 | 8131156 | 0 | 0 | 0 | 8131156 |
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.