A number of the form (n3 + n)/2. more
The first 600 magic square constants :
1,
5,
15,
34,
65,
111,
175,
260,
369,
505,
671,
870,
1105,
1379,
1695,
2056,
2465,
2925,
3439,
4010,
4641,
5335,
6095,
6924,
7825,
8801,
9855,
10990,
12209,
13515,
14911,
16400,
17985,
19669,
21455,
23346,
25345,
27455,
29679,
32020,
34481,
37065,
39775,
42614,
45585,
48691,
51935,
55320,
58849,
62525,
66351,
70330,
74465,
78759,
83215,
87836,
92625,
97585,
102719,
108030,
113521,
119195,
125055,
131104,
137345,
143781,
150415,
157250,
164289,
171535,
178991,
186660,
194545,
202649,
210975,
219526,
228305,
237315,
246559,
256040,
265761,
275725,
285935,
296394,
307105,
318071,
329295,
340780,
352529,
364545,
376831,
389390,
402225,
415339,
428735,
442416,
456385,
470645,
485199,
500050,
515201,
530655,
546415,
562484,
578865,
595561,
612575,
629910,
647569,
665555,
683871,
702520,
721505,
740829,
760495,
780506,
800865,
821575,
842639,
864060,
885841,
907985,
930495,
953374,
976625,
1000251,
1024255,
1048640,
1073409,
1098565,
1124111,
1150050,
1176385,
1203119,
1230255,
1257796,
1285745,
1314105,
1342879,
1372070,
1401681,
1431715,
1462175,
1493064,
1524385,
1556141,
1588335,
1620970,
1654049,
1687575,
1721551,
1755980,
1790865,
1826209,
1862015,
1898286,
1935025,
1972235,
2009919,
2048080,
2086721,
2125845,
2165455,
2205554,
2246145,
2287231,
2328815,
2370900,
2413489,
2456585,
2500191,
2544310,
2588945,
2634099,
2679775,
2725976,
2772705,
2819965,
2867759,
2916090,
2964961,
3014375,
3064335,
3114844,
3165905,
3217521,
3269695,
3322430,
3375729,
3429595,
3484031,
3539040,
3594625,
3650789,
3707535,
3764866,
3822785,
3881295,
3940399,
4000100,
4060401,
4121305,
4182815,
4244934,
4307665,
4371011,
4434975,
4499560,
4564769,
4630605,
4697071,
4764170,
4831905,
4900279,
4969295,
5038956,
5109265,
5180225,
5251839,
5324110,
5397041,
5470635,
5544895,
5619824,
5695425,
5771701,
5848655,
5926290,
6004609,
6083615,
6163311,
6243700,
6324785,
6406569,
6489055,
6572246,
6656145,
6740755,
6826079,
6912120,
6998881,
7086365,
7174575,
7263514,
7353185,
7443591,
7534735,
7626620,
7719249,
7812625,
7906751,
8001630,
8097265,
8193659,
8290815,
8388736,
8487425,
8586885,
8687119,
8788130,
8889921,
8992495,
9095855,
9200004,
9304945,
9410681,
9517215,
9624550,
9732689,
9841635,
9951391,
10061960,
10173345,
10285549,
10398575,
10512426,
10627105,
10742615,
10858959,
10976140,
11094161,
11213025,
11332735,
11453294,
11574705,
11696971,
11820095,
11944080,
12068929,
12194645,
12321231,
12448690,
12577025,
12706239,
12836335,
12967316,
13099185,
13231945,
13365599,
13500150,
13635601,
13771955,
13909215,
14047384,
14186465,
14326461,
14467375,
14609210,
14751969,
14895655,
15040271,
15185820,
15332305,
15479729,
15628095,
15777406,
15927665,
16078875,
16231039,
16384160,
16538241,
16693285,
16849295,
17006274,
17164225,
17323151,
17483055,
17643940,
17805809,
17968665,
18132511,
18297350,
18463185,
18630019,
18797855,
18966696,
19136545,
19307405,
19479279,
19652170,
19826081,
20001015,
20176975,
20353964,
20531985,
20711041,
20891135,
21072270,
21254449,
21437675,
21621951,
21807280,
21993665,
22181109,
22369615,
22559186,
22749825,
22941535,
23134319,
23328180,
23523121,
23719145,
23916255,
24114454,
24313745,
24514131,
24715615,
24918200,
25121889,
25326685,
25532591,
25739610,
25947745,
26156999,
26367375,
26578876,
26791505,
27005265,
27220159,
27436190,
27653361,
27871675,
28091135,
28311744,
28533505,
28756421,
28980495,
29205730,
29432129,
29659695,
29888431,
30118340,
30349425,
30581689,
30815135,
31049766,
31285585,
31522595,
31760799,
32000200,
32240801,
32482605,
32725615,
32969834,
33215265,
33461911,
33709775,
33958860,
34209169,
34460705,
34713471,
34967470,
35222705,
35479179,
35736895,
35995856,
36256065,
36517525,
36780239,
37044210,
37309441,
37575935,
37843695,
38112724,
38383025,
38654601,
38927455,
39201590,
39477009,
39753715,
40031711,
40311000,
40591585,
40873469,
41156655,
41441146,
41726945,
42014055,
42302479,
42592220,
42883281,
43175665,
43469375,
43764414,
44060785,
44358491,
44657535,
44957920,
45259649,
45562725,
45867151,
46172930,
46480065,
46788559,
47098415,
47409636,
47722225,
48036185,
48351519,
48668230,
48986321,
49305795,
49626655,
49948904,
50272545,
50597581,
50924015,
51251850,
51581089,
51911735,
52243791,
52577260,
52912145,
53248449,
53586175,
53925326,
54265905,
54607915,
54951359,
55296240,
55642561,
55990325,
56339535,
56690194,
57042305,
57395871,
57750895,
58107380,
58465329,
58824745,
59185631,
59547990,
59911825,
60277139,
60643935,
61012216,
61381985,
61753245,
62125999,
62500250,
62876001,
63253255,
63632015,
64012284,
64394065,
64777361,
65162175,
65548510,
65936369,
66325755,
66716671,
67109120,
67503105,
67898629,
68295695,
68694306,
69094465,
69496175,
69899439,
70304260,
70710641,
71118585,
71528095,
71939174,
72351825,
72766051,
73181855,
73599240,
74018209,
74438765,
74860911,
75284650,
75709985,
76136919,
76565455,
76995596,
77427345,
77860705,
78295679,
78732270,
79170481,
79610315,
80051775,
80494864,
80939585,
81385941,
81833935,
82283570,
82734849,
83187775,
83642351,
84098580,
84556465,
85016009,
85477215,
85940086,
86404625,
86870835,
87338719,
87808280,
88279521,
88752445,
89227055,
89703354,
90181345,
90661031,
91142415,
91625500,
92110289,
92596785,
93084991,
93574910,
94066545,
94559899,
95054975,
95551776,
96050305,
96550565,
97052559,
97556290,
98061761,
98568975,
99077935,
99588644,
100101105,
100615321,
101131295,
101649030,
102168529,
102689795,
103212831,
103737640,
104264225,
104792589,
105322735,
105854666,
106388385,
106923895,
107461199,
108000300.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 125992 values, from 1 to 999997500158740).
n\r | 0 | 1 |
2 | 31498 | 94494 | 2 |
3 | 41997 | 41998 | 41997 | 3 |
4 | 15749 | 47247 | 15749 | 47247 | 4 |
5 | 75595 | 25199 | 0 | 0 | 25198 | 5 |
6 | 10499 | 31498 | 10499 | 31498 | 10500 | 31498 | 6 |
7 | 17998 | 35998 | 17999 | 0 | 0 | 17999 | 35998 | 7 |
8 | 7874 | 39372 | 7875 | 7874 | 7875 | 7875 | 7874 | 39373 | 8 |
9 | 13999 | 14000 | 13999 | 13999 | 13999 | 13999 | 13999 | 13999 | 13999 | 9 |
10 | 18899 | 18900 | 0 | 0 | 6300 | 56696 | 6299 | 0 | 0 | 18898 | 10 |
11 | 11453 | 34362 | 0 | 0 | 11454 | 11454 | 11454 | 11454 | 0 | 0 | 34361 |
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.