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magic square constants
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 
23149894494 2 
3419974199841997 3 
415749472471574947247 4 
575595251990025198 5 
6104993149810499314981050031498 6 
7179983599817999001799935998 7 
87874393727875787478757875787439373 8 
9139991400013999139991399913999139991399913999 9 
1018899189000063005669662990018898 10 
11114533436200114541145411454114540034361

A pictorial representation of the table above
motab
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.