410
The smallest squares containing k 410's :
44100 = 2102,
6514104100 = 807102,
1941034104100 = 13932102.
The square root of 410 is 20.24..., 20 = 22 + 42.
190k + 314k + 410k + 530k are squares for k = 1,2,3 (382, 7642, 159882).
4102 = 168100, 16 = 42 and 8100 = 902.
4102 = 19 x 20 + 21 x 22 + 23 x 24 + 25 x 26 + ... + 99 x 100.
4102 = (12 + 1)(92 + 1)(322 + 1).
(13 + 23 + ... + 1673)(1683 + 1693 + ... + 4103) = 11654322122.
4102 = 168100 appears in the decimal expression of e:
e = 2.71828•••168100••• (from the 9802nd digit),
(168100 is the eighth 6-digit square in the expression of e.)
by Yoshio Mimura, Kobe, Japan
411
The smallest squares containing k 411's :
1094116 = 10462,
1641141121 = 405112,
411141156411456 = 202766162.
The square root of 411 is 20.273134932713...,
202 = 22 + 72 + 32 + 12 + 32 + 42 + 92 + 32 + 22 + 72 + 132.
4112 = 168921, 1 * 6 * 8 * 9 - 21 = 411.
1233k + 27948k + 55074k + 84666k are squares for k = 1,2,3 (4112, 1048052, 282098072).
2329k + 38908k + 53978k + 73706k are squares for k = 1,2,3 (4112, 993252, 248313872).
3-by-3 magic squares consisting of different squares with constant 4112:
| A2 | B2 | C2 |
| D2 | E2 | F2 |
| G2 | H2 | K2 |
| where (A, B, C, D, E, F, G, H, K) = | |
| (10, 265, 314, 286, 230, 185, 295, 214, 190), | (14, 151, 382, 226, 322, 119, 343, 206, 94), |
| (34, 199, 358, 263, 286, 134, 314, 218, 151), | (36, 171, 372, 288, 276, 99, 291, 252, 144), |
| (41, 94, 398, 146, 377, 74, 382, 134, 71), | (41, 218, 346, 262, 281, 146, 314, 206, 167), |
| (46, 217, 346, 242, 266, 199, 329, 226, 98), | (74, 182, 361, 214, 329, 122, 343, 166, 154) |
4112 = 168921, 1 + 6 + 892 + 1 = 302,
4112 = 168921, 1 + 68 + 9 + 2 + 1 = 92,
4112 = 168921, 16 + 8 + 9 + 2 + 1 = 62,
4112 = 168921, 168 + 921 = 332.
by Yoshio Mimura, Kobe, Japan
412
The smallest squares containing k 412's :
41209 = 2032,
1341244129 = 366232,
64124124124516 = 80077542.
4122 = 44 + 144 + 164 + 164.
16789k + 25647k + 55517k + 71791k are squares for k = 1,2,3 (4122, 957902, 237217242).
4122 = 169744, 1 + 69 + 7 + 44 = 112,
4122 = 169744, 16 + 97 + 4 + 4 = 112.
by Yoshio Mimura, Kobe, Japan
413
The smallest squares containing k 413's :
141376 = 3762,
41398413156 = 2034662,
141341354131849 = 118887072.
4132 = 170569, a square with different digits.
4132 = 222 + 232 + 242 + ... + 802.
4132 = 170569 with 1 = 12, 7056 = 842 and 9 = 32.
34692k + 38409k + 45430k + 52038k are squares for k = 1,2,3 (4132, 863172, 182508832).
Komachi Square Sum : 4132 = 82 + 942 + 1752 + 3622.
(156 / 413)2 = 0.142675398... , (221 / 413)2 = 0.286341597... (Komachic).
12 + 22 + ... + 4132 = 23567019, which consists of different digits (the third 8-digit sum).
3-by-3 magic squares consisting of different squares with constant 4132:
| A2 | B2 | C2 |
| D2 | E2 | F2 |
| G2 | H2 | K2 |
| where (A, B, C, D, E, F, G, H, K) = | |
| (3, 64, 408, 216, 348, 53, 352, 213, 36), | (3, 152, 384, 288, 276, 107, 296, 267, 108), |
| (8, 117, 396, 144, 372, 107, 387, 136, 48), | (12, 141, 388, 179, 348, 132, 372, 172, 51), |
| (12, 244, 333, 276, 243, 188, 307, 228, 156), | (44, 192, 363, 267, 264, 172, 312, 253, 96) |
4132 + 4142 + 4152 + 4162 + ... + 12992 = 266102.
4132 + 4142 + 4152 + 4162 + ... + 1292932 = 268413532.
by Yoshio Mimura, Kobe, Japan
414
The smallest squares containing k 414's :
414736 = 6442,
4144140625 = 643752,
3041494144144 = 17439882.
4142 = 171396, 17 + 1 + 396 = 414.
4142 = 171396, a square with odd digits except the last digit 6.
4142 = (12 + 5)(1692 + 5) = (14 + 5)(134 + 5).
10626k + 33258k + 36570k + 90942k are squares for k = 1,2,3 (4142, 1040522, 289659242).
15042k + 16698k + 49818k + 89838k are squares for k = 1,2,3 (4142, 1051562, 292706282).
23874k + 33258k + 39054k + 75210k are squares for k = 1,2,3 (4142, 941162, 231384602).
The 4-by-4 magic squares consisting of different squares with constant 414:
|
|
4142 = 171396, 1 + 7 + 13 + 9 + 6 = 62,
4142 = 171396, 1 + 713 + 9 + 6 = 272,
4142 = 171396, 17 + 1 + 3 + 9 + 6 = 62.
by Yoshio Mimura, Kobe, Japan
415
The smallest squares containing k 415's :
3041536 = 17442,
4153415809 = 644472,
1841514154159689 = 429128672.
4152 = 343 + 403 + 413.
(342 / 415)2 = 0.679134852... (Komachic).
3-by-3 magic squares consisting of different squares with constant 4152:
| A2 | B2 | C2 |
| D2 | E2 | F2 |
| G2 | H2 | K2 |
| where (A, B, C, D, E, F, G, H, K) = | |
| (6, 235, 342, 258, 270, 181, 325, 210, 150), | (10, 90, 405, 171, 370, 78, 378, 165, 46), |
| (10, 90, 405, 270, 309, 62, 315, 262, 66), | (21, 170, 378, 222, 315, 154, 350, 210, 75), |
| (27, 150, 386, 186, 350, 123, 370, 165, 90), | (53, 210, 354, 246, 270, 197, 330, 235, 90), |
| (66, 190, 363, 213, 330, 134, 350, 165, 150) |
4152 = 172225, 1 + 72 + 2 + 25 = 102,
4152 = 172225, 1 + 72 + 22 + 5 = 102,
4152 = 172225, 17 + 22 + 25 = 82.
by Yoshio Mimura, Kobe, Japan
416
The smallest squares containing k 416's :
14161 = 1192,
416241604 = 204022,
34165416124161 = 58451192.
The squares which begin with 416 and end in 416 are
41696006416 = 2041962, 416277878416 = 6451962, 416417252416 = 6453042,
416923324416 = 6456962, 4160359772416 = 20396962,...
4162 = 173056, a square with different digits.
4162± 3 are primes.
4162 = 173056, 1 + 7 + 30 + 5 + 6 = 72.
Page of Squares : First Upload February 7, 2005 ; Last Revised January 16, 2014by Yoshio Mimura, Kobe, Japan
417
The smallest squares containing k 417's :
417316 = 6462,
4177941769 = 646372,
364178941741729 = 190834732.
The square root of 417 is 20.42..., 20 = 42 + 22.
4172 = 84 + 124 + 164 + 174.
138k + 417k + 582k + 888k are squares for k = 1,2,3 (452, 11492, 311852).
13761k + 36696k + 44202k + 79230k are squares for k = 1,2,3 (4172, 988292, 252139052).
4173 = 72511713 : 72 + 22 + 52 + 112 + 72 + 132 = 417.
3-by-3 magic squares consisting of different squares with constant 4172:
| A2 | B2 | C2 |
| D2 | E2 | F2 |
| G2 | H2 | K2 |
| where (A, B, C, D, E, F, G, H, K) = | |
| (7, 64, 412, 272, 313, 44, 316, 268, 47), | (8, 103, 404, 217, 344, 92, 356, 212, 47), |
| (8, 212, 359, 281, 268, 152, 308, 239, 148), | (16, 223, 352, 257, 272, 184, 328, 224, 127), |
| (27, 198, 366, 282, 261, 162, 306, 258, 117), | (28, 176, 377, 244, 313, 128, 337, 212, 124), |
| (40, 167, 380, 292, 260, 145, 295, 280, 92), | (64, 127, 392, 152, 376, 97, 383, 128, 104), |
| (68, 167, 376, 236, 328, 103, 337, 196, 148), | (71, 212, 352, 272, 292, 121, 308, 209, 188) |
4172 = 173889, 1 + 7 + 3 + 8 + 8 + 9 = 62,
4172 = 173889, 1 + 7 + 3 + 889 = 302,
4172 = 173889, 17 + 38 + 89 = 122.
by Yoshio Mimura, Kobe, Japan
418
The smallest squares containing k 418's :
418609 = 6472,
11418418449 = 1068572,
341842418418361 = 184889812.
4182 = 174724, a zigzag square.
7106k + 33022k + 40546k + 94050k are squares for k = 1,2,3 (4182, 1078442, 305767002).
21945k + 33649k + 44517k + 74613k are squares for k = 1,2,3 (4182, 957222, 235003782).
Komachi equations:
4182 = - 92 * 82 * 72 + 6542 + 322 * 12 = - 92 * 82 * 72 + 6542 + 322 / 12,
4182 = 93 + 83 * 73 - 63 - 53 - 43 - 33 * 23 - 103.
4182 = 174724, 1 + 7 + 4 + 7 + 2 + 4 = 52,
4182 = 174724, 1 + 7 + 472 + 4 = 222.
by Yoshio Mimura, Kobe, Japan
419
The smallest squares containing k 419's :
264196 = 5142,
41964341904 = 2048522,
419041943424196 = 204705142.
4192 = 175561, 17 * 5 * 5 - 6 * 1 = 419.
22k + 161k + 266k + 280k are squares for k = 1,2,3 (272, 4192, 67052).
3-by-3 magic squares consisting of different squares with constant 4192:
| A2 | B2 | C2 |
| D2 | E2 | F2 |
| G2 | H2 | K2 |
| where (A, B, C, D, E, F, G, H, K) = | |
| (14, 63, 414, 198, 366, 49, 369, 194, 42), | (31, 114, 402, 222, 346, 81, 354, 207, 86), |
| (33, 186, 374, 234, 319, 138, 346, 198, 129), | (49, 174, 378, 282, 266, 159, 306, 273, 86), |
| (66, 159, 382, 193, 354, 114, 366, 158, 129) |
4192 = 175561, 1 + 7 + 5 + 5 + 6 + 1 = 52.
Page of Squares : First Upload February 7, 2005 ; Last Revised March 4, 2011by Yoshio Mimura, Kobe, Japan