400
The smallest squares containing k 400's :
400 = 202,
4004001 = 20012,
240080400400 = 4899802.
The squares which begin with 400 and end in 400 are
400800400 = 200202, 4004358400 = 632802, 4009422400 = 633202,
40008000400 = 2000202, 40032006400 = 2000802,...
202 = 1 + 7 + 72 + 73.
202 = (12 + 4)(22 + 4)(42 + 4)(142 + 4) = (42 + 4)(62 + 4)(142 + 4).
Komachi equations:
4002 = - 12 * 22 + 342 + 562 * 72 + 82 * 92,
4002 = 94 * 84 / 74 / 64 / 544 * 34 * 2104.
202 = 203 + 303 + 503.
4004001 = 20012.
Page of Squares : First Upload anuary 31, 2005 ; Last Revised December 7, 2013by Yoshio Mimura, Kobe, Japan
401
The smallest squares containing k 401's :
2401 = 492,
401401225 = 200352,
104010401705401 = 101985492.
The squares which begin with 401 and end in 401 are
401962401 = 200492, 40119689401 = 2002992, 40180603401 = 2004512,
401067623401 = 6332992, 401260169401 = 6334512,...
The square root of 401 is 20.024..., 20 = 02 + 22 + 42.
4012 = 160801, a zigzag square.
Cubic Polynomial : (X + 872)(X + 1442)(X + 3642) = X3 + 4012X2 + 625082X + 45601922.
Komachi equations:
4012 = 92 - 82 + 762 * 52 + 42 * 322 * 12 = 92 - 82 + 762 * 52 + 42 * 322 / 12.
3-by-3 magic squares consisting of different squares with constant 4012:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(4, 63, 396, 279, 284, 48, 288, 276, 41), | (24, 140, 375, 265, 276, 120, 300, 255, 76), |
(24, 183, 356, 252, 284, 129, 311, 216, 132), | (41, 108, 384, 144, 364, 87, 372, 129, 76), |
(84, 228, 319, 256, 279, 132, 297, 176, 204) |
4012 = 160801, 1 + 6 + 0 + 8 + 0 + 1 = 42,
4012 = 160801, 16 + 0 + 8 + 0 + 1 = 52,
4012 = 160801, 160 + 8 + 0 + 1 = 132,
4012 = 160801, 160 + 801 = 312.
by Yoshio Mimura, Kobe, Japan
402
The smallest squares containing k 402's :
24025 = 1552,
4020194025 = 634052,
180402640274025 = 134314052.
4022 = 161604, a zigzag square.
Komachi equation: 4022 = - 12 + 22 * 32 - 42 + 562 * 72 + 892.
The 4-by-4 magic square consisting of different squares with constant 402:
|
4022 = 161604, 16 + 1 + 60 + 4 = 92,
4022 = 161604, 16 + 16 + 0 + 4 = 62,
4022 = 161604, 161 + 60 + 4 = 152.
46202 = 1062 + 1072 + 1082 + 1092 + ... + 4022.
Page of Squares : First Upload anuary 31, 2005 ; Last Revised June 4, 2010by Yoshio Mimura, Kobe, Japan
403
The smallest squares containing k 403's :
244036 = 4942,
6403040361 = 800192,
403940336403289 = 200982672.
403 = (12 + 22 + 32 + ... + 772) / (12 + 22 + 32 + ... + 102).
4032 = 162409, 1 + 6 + 240 + 9 = 162,
4032 = 162409, 16 + 24 + 0 + 9 = 72.
4032 = 162409, a zigzag square with different digits.
403k + 961k + 5363k + 8649k are squares for k = 1,2,3 (1242, 102302, 8956522).
3-by-3 magic squares consisting of different squares with constant 4032:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(2, 162, 369, 198, 321, 142, 351, 182, 78), | (6, 63, 398, 177, 358, 54, 362, 174, 33), |
(30, 97, 390, 222, 330, 65, 335, 210, 78), | (47, 162, 366, 258, 294, 97, 306, 223, 138), |
(78, 159, 362, 182, 342, 111, 351, 142, 138), | (78, 254, 303, 273, 258, 146, 286, 177, 222) |
4032 + 4042 + 4052 + 4062 + ... + 295542 = 29334202.
Page of Squares : First Upload anuary 31, 2005 ; Last Revised March 4, 2011by Yoshio Mimura, Kobe, Japan
404
The smallest squares containing k 404's :
10404 = 1022,
404090404 = 201022,
194404404638404 = 139428982.
The squares which begin with 404 and end in 404 are
404090404 = 201022, 4045214404 = 636022, 40442014404 = 2011022,
404366266404 = 6358982, 404625754404 = 6361022,...
The square root of 404 is 20.099751242...,
and 202 = 02 + 92 + 92 + 72 + 52 + 122 + 42 + 22.
4042 = 163216, 16 + 32 + 16 = 82.
Page of Squares : First Upload anuary 31, 2005 ; Last Revised July 13, 2006by Yoshio Mimura, Kobe, Japan
405
The smallest squares containing k 405's :
405769 = 6372,
8405405761 = 916812,
974054050405921 = 312098392.
The square root of 405 is 20.1246117974981...,
202 = 12 + 22 + 42 + 62 + 12 + 12 + 72 + 92 + 72 + 42 + 92 + 82 + 12.
4052 = 164025, a square with different digits.
4052 = 93 + 183 + 543.
4052 = (23 + 24 + 25 + ... + 31)2 + (32 + 33 + 34 + ... + 40)2.
Loop of length 56 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
405 - 41 - 1681 - 6817 - ... - 7684 - 12832 - 1809 - 405
(Note f(405) = 42 + 052 = 41, f(41) = 412 = 1681, etc. See 41)
Komachi Cubic Sum : 4052 = 93 + 183 + 273 + 363 + 453.
12 + 22 + ... + 4052 = 22225455, which consists of 3 kinds of digits.
3-by-3 magic squares consisting of different squares with constant 4052:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(5, 116, 388, 220, 325, 100, 340, 212, 59), | (5, 140, 380, 220, 320, 115, 340, 205, 80), |
(11, 200, 352, 248, 275, 164, 320, 220, 115), | (16, 88, 395, 115, 380, 80, 388, 109, 40), |
(28, 160, 371, 275, 280, 100, 296, 245, 128), | (30, 105, 390, 150, 366, 87, 375, 138, 66), |
(30, 150, 375, 210, 327, 114, 345, 186, 102), | (40, 173, 364, 245, 280, 160, 320, 236, 77), |
(40, 245, 320, 268, 224, 205, 301, 232, 140), | (80, 205, 340, 235, 304, 128, 320, 172, 179), |
(80, 235, 320, 269, 208, 220, 292, 256, 115) |
4052 = 164025, 1 + 6 + 4 + 0 + 25 = 62,
4052 = 164025, 16 + 40 + 25 = 92.
(13 + 23 + ... + 3143)(3153 + 3163 + ... + 3503)(3513 + 3523 + ... + 4053) = 984575065320002.
Page of Squares : First Upload anuary 31, 2005 ; Last Revised April 6, 2009by Yoshio Mimura, Kobe, Japan
406
The smallest squares containing k 406's :
140625 = 3752,
40640625 = 63752,
9024064064064 = 30040082.
4062 = 164836, a zigzag square.
4062 + 4072 + 4082 + ... + 4202 = 4212 + 4222 + 4232 + ... + 4342.
11165k + 24157k + 46893k + 82621k are squares for k = 1,2,3 (4062, 986582, 261265062).
4062 = 164836, 1 + 6 + 4 + 83 + 6 = 102,
4062 = 164836, 1 + 6 + 48 + 3 + 6 = 82,
4062 = 164836, 12 + 6482 + 362 = 6492,
4062 = 164836, 16 + 4 + 8 + 36 = 82,
4062 = 164836, 16 + 48 + 36 = 102.
(13 + 23 + ... + 2313)(2323 + 2333 + ... + 4063) = 20942413802.
Page of Squares : First Upload anuary 31, 2005 ; Last Revised September 6, 2011by Yoshio Mimura, Kobe, Japan
407
The smallest squares containing k 407's :
407044 = 6382,
3407407129 = 583732,
407873407407376 = 201958762.
The square root of 407 is 20.1742410018320143988...,
202 = 12 + 72 + 42 + 22 + 42 + ... + 42 + 32 + 92 + 82 + 82.
407 = (12 + 22 + 32 + ... + 552) / (12 + 22 + 32 + ... + 72).
The square root of 407 is 20.17424100183...,
202 = 172 + 42 + 22 + 42 + 12 + 02 + 02 + 12 + 82 + 32.
4072 = 165649, a zigzag square.
The sum of (19x + 12)2 for x = 0,1,2,...,10 is 4072.
A + B, A + C, A + D, B + C, B + D, C + D are squares for A = 407, B = 3314, C = 4082, D = 5522.
253k + 295k + 341k + 407k are squares for k = 1,2,3 (362, 6582, 122042).
3-by-3 magic squares consisting of different squares with constant 4072:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(2, 126, 387, 234, 317, 102, 333, 222, 74), | (3, 102, 394, 178, 354, 93, 366, 173, 42), |
(3, 186, 362, 254, 282, 147, 318, 227, 114), | (18, 61, 402, 142, 378, 51, 381, 138, 38), |
(18, 115, 390, 285, 282, 70, 290, 270, 93), | (45, 250, 318, 282, 210, 205, 290, 243, 150), |
(74, 222, 333, 243, 294, 142, 318, 173, 186) |
4072 = 165649, 1 + 65 + 6 + 49 = 112,
4072 = 165649, 16 + 56 + 49 = 112.
by Yoshio Mimura, Kobe, Japan
408
The smallest squares containing k 408's :
40804 = 2022,
20408408164 = 1428582,
264084080440896 = 162506642.
4082 = 166464, a square with 3 kinds of digits.
4082 = 166464, 1 * 6 * 64 + 6 * 4 = 16 * 6 * 4 + 6 * 4 = 408.
4082 = (32 + 8)(42 + 8)(202 + 8) = (82 + 8)(482 + 8).
408k + 1156k + 2601k + 3060k are squares for k = 1,2,3 (852, 41992, 2187732).
6k + 408k + 690k + 921k are squares for k = 1,2,3 (452, 12212, 343172).
4082 = 166464, 1 + 6 + 6 + 4 + 64 = 92,
4082 = 166464, 1 + 6 + 64 + 6 + 4 = 92,
4082 = 166464, 1 + 66 + 4 + 6 + 4 = 92,
4082 = 166464, 1 + 664 + 64 = 272,
4082 = 166464, 16 + 6 + 4 + 6 + 4 = 62,
4082 = 166464, 16 + 64 + 64 = 122.
4082 = 166464, with 16 = 42 and 64 = 82.
4082 = 103 + 203 + 543 = 103 + 383 + 483.
4082 = 166464 is an exchangeable square, 646416 = 8042.
4082 = 2 x 3 x 4 + 4 x 5 x 6 + 6 x 7 x 8 + ... + 32 x 33 x 34.
Page of Squares : First Upload anuary 31, 2005 ; Last Revised December 7, 2013by Yoshio Mimura, Kobe, Japan
409
The smallest squares containing k 409's :
4096 = 642,
40998409 = 64032,
8409437409409 = 28999032.
The squares which begin with 409 and end in 409 are
40998409 = 64032, 40944308409 = 2023472, 40966974409 = 2024032,
409084322409 = 6395972, 409155960409 = 6396532,...
4092 + 4102 + 4112 + ... + 2262242 = 621224862.
409 is the seventh prime for which the Legendre Symbol (a/409) = 1 for a = 1, 2,,..,6.
(12 + 22 + ... + 1892) + (12 + 22 + ... + 3952) = (12 + 22 + ... + 4092).
3-by-3 magic squares consisting of different squares with constant 4092:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
(12, 176, 369, 201, 324, 148, 356, 177, 96), | (36, 111, 392, 183, 356, 84, 364, 168, 81) |
(49, 132, 384, 228, 329, 84, 336, 204, 113), | (76, 201, 348, 264, 292, 111, 303, 204, 184) |
4092 = 167281, 1 + 6 + 7 + 2 + 8 + 1 = 52,
4092 = 167281, 16 + 72 + 81 = 132,
4092 = 167281, 167 + 28 + 1 = 142,
4092 = 167281, 1672 + 8 + 1 = 412.
(13 + 23 + ... + 1203)(1213 + 1223 + ... + 1643)(1653 + 1663 + ... + 4093) = 68587561350002.
Page of Squares : First Upload anuary 31, 2005 ; Last Revised april 6, 2009by Yoshio Mimura, Kobe, Japan