690
The smallest squares containing k 690's :
16900 = 1302,
690690961 = 262812,
669069096976900 = 258663702.
The sum of (11x+3)2 is 6902, the sum is taken for x = 0, 1, 2,..., 22.
The sum of the cubes of the divisors of 690 is 196562.
6902 is the first square which is the sum of 6 eighth powers : 28 + 38 + 38 + 38 + 48 + 58.
6902 = (52 + 5)(82 + 5)(152 + 5).
Loop of length 56 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
690 - 8136 - 7857 - 9333 - ... - 7450 - 7976 - 12017 - 690
(Note f(690) = 62 + 902 = 8136, f(8136) = 812 + 362 = 7857, etc. See 41)
1 / 690 = 0.00144..., where 144 = 122.
510k + 690k + 1330k + 9570k are squares for k = 1,2,3 (1102, 97002, 9377002).
6k + 408k + 690k + 921k are squares for k = 1,2,3 (452, 12212, 343172).
Komachi equation: 6902 = - 93 * 83 + 763 * 53 / 43 - 33 - 23 * 103.
The 4-by-4 magic squares consisting of different squares with constant 690:
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6902 = 476100 appears in the decimal expression of π:
π = 3.14159•••476100••• (from the 30316th digit).
by Yoshio Mimura, Kobe, Japan
691
The smallest squares containing k 691's :
69169 = 2632,
6916916224 = 831682,
271691069166916 = 164830542.
(12 + 22 + ... + 112)(122 + 132 + ... + 1152)(1162 + 1172 + ... + 6912) = 1687651682.
1 / 691 = 0.00144..., where 144 = 122.
The square root of 691 is 26. 2 8 6 8 7 8 8 5 6 1 8 9 8 3 1 0 4 3 0 5 ...
and 262 = 22 + 82 + 62 + 82 + 72 + 82 + 82 + 52 + 62 + 12 + 82 + 92 + 82 + 32 + 12 + 02 + 42 + 32 + 02 + 52.
3-by-3 magic squares consisting of different squares with constant 6912:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(6, 214, 657, 358, 561, 186, 591, 342, 106), | (6, 223, 654, 258, 606, 209, 641, 246, 78), |
(6, 353, 594, 479, 426, 258, 498, 414, 241), | (18, 81, 686, 326, 606, 63, 609, 322, 54), |
(34, 129, 678, 426, 538, 81, 543, 414, 106), | (54, 294, 623, 462, 479, 186, 511, 402, 234), |
(63, 214, 654, 434, 522, 129, 534, 399, 182), | (74, 318, 609, 417, 466, 294, 546, 399, 142), |
(111, 414, 542, 466, 447, 246, 498, 326, 351) |
Page of Squares : First Upload August 18, 2005 ; Last Revised July 27, 2009
by Yoshio Mimura, Kobe, Japan
692
The smallest squares containing k 692's :
106929 = 3272,
16924969216 = 1300962,
692692761369249 = 263190572.
(387 / 692)2 = 0.312758946... (Komachic).
6922 + 6932 + 6942 + ... + 15552 = 338282.
1 / 692 = 0.00144..., and 144 = 122.
6922 = 478864, 4 + 78 * 8 + 64 = 692.
6922 = 63 + 163 + 783.
210k + 241k + 538k + 692k are squares for k = 1,2,3 (412, 9332, 225912).
6922 = 478864, 4 + 78 + 8 + 6 + 4 = 102.
Page of Squares : First Upload August 18, 2005 ; Last Revised March 23, 2011by Yoshio Mimura, Kobe, Japan
693
The smallest squares containing k 693's :
269361 = 5192,
16930693924 = 1301182,
16936936933401 = 41154512.
Loop of length 35 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
693 - 8685 - 14621 - 2558 - ... - 3394 - 9925 - 10426 - 693
(Note f(693) = 62 + 932 = 8685, f(8685) = 862 + 852 = 14621, etc. See 37)
(634 / 693)2 = 0.836974152... (Komachic).
1 / 693 = 0.00144..., and 144 = 122.
6932 = (53 + 54 + 55)2 + (56 + 57 + 58)2 + (59 + 60 + 61)2 + ... + (83 + 84 + 85)2.
6932 = 2312 + 4622 + 4622 : 2642 + 2642 + 1322 = 3962.
Cubic Polynomial :
(X + 1962)(X + 5282)(X + 6932) = X3 + 8932X2 + 4037882X + 717171842.
(1 + 2)(3 + 4)(5 + 6)(7)(8 + 9 + 10)(11) = 6932,
(1 + 2 + ... + 6)(7 + 8 + ... + 15)(16 + 17 + ... + 26) = 6932,
(1 + 2 + ... + 98)(99) = 6932.
The square root of 693 is 26. 3 2 4 8 9 3 16 2 1 7 6 3 6 6 1 8 1 ...,
and 262 = 32 + 22 + 42 + 82 + 92 + 32 + 162 + 22 + 12 + 72 + 62 + 32 + 62 + 62 + 12 + 82 + 12.
12 + 22 + 32 + ... + 6932 = 111177759, which consists of odd digits (the first 9-digit),
there are two 9-digit sums consisting of odd digits (cf. 981).
3-by-3 magic squares consisting of different squares with constant 6932:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(4, 83, 688, 467, 508, 64, 512, 464, 53), | (4, 188, 667, 293, 604, 172, 628, 283, 76), |
(4, 232, 653, 397, 536, 188, 568, 373, 136), | (6, 282, 633, 438, 489, 222, 537, 402, 174), |
(8, 373, 584, 404, 472, 307, 563, 344, 212), | (13, 152, 676, 424, 533, 128, 548, 416, 83), |
(13, 184, 668, 284, 608, 173, 632, 277, 64), | (14, 217, 658, 343, 574, 182, 602, 322, 119), |
(16, 368, 587, 412, 467, 304, 557, 356, 208), | (22, 374, 583, 473, 418, 286, 506, 407, 242), |
(30, 318, 615, 375, 510, 282, 582, 345, 150), | (32, 212, 659, 488, 461, 172, 491, 472, 128), |
(37, 412, 556, 452, 436, 293, 524, 347, 292), | (39, 138, 678, 258, 633, 114, 642, 246, 87), |
(40, 307, 620, 395, 520, 232, 568, 340, 205), | (52, 229, 652, 421, 508, 212, 548, 412, 101), |
(57, 222, 654, 438, 519, 138, 534, 402, 183), | (68, 316, 613, 347, 548, 244, 596, 283, 212), |
(78, 249, 642, 327, 582, 186, 606, 282, 183), | (87, 366, 582, 474, 393, 318, 498, 438, 201), |
(92, 269, 632, 472, 488, 139, 499, 412, 248), | (98, 329, 602, 406, 518, 217, 553, 322, 266), |
(102, 426, 537, 471, 438, 258, 498, 327, 354), | (107, 344, 592, 472, 397, 316, 496, 452, 173), |
(109, 208, 652, 272, 619, 152, 628, 232, 179), | (115, 260, 632, 380, 557, 160, 568, 320, 235), |
(148, 376, 563, 403, 512, 236, 544, 277, 328), | (149, 328, 592, 368, 548, 211, 568, 269, 292) |
6932 = 480249, 43 + 83 + 03 + 243 + 93 = 1232,
6932 = 480249, 48 + 0 + 24 + 9 = 92,
6932 = 480249, 480 + 249 = 272.
6932 = 480249 appears in the decimal expression of π:
π = 3.14159•••480249••• (from the 27974th digit).
by Yoshio Mimura, Kobe, Japan
694
The smallest squares containing k 694's :
506944 = 7122,
3694694656 = 607842,
26948069469409 = 51911532.
1 / 694 = 0.00144..., and 144 = 122.
6942 = 481636, a zigzag square.
6942± 3 are primes.
6942 = 481636 is an exchangeable square, 364816 = 6042.
6942 + 6952 + 6962 + ... + 19502 = 486042.
6942 = 481636, 4 + 8 + 16 + 36 = 82,
6942 = 481636, 4 + 81 + 6 + 3 + 6 = 102,
6942 = 481636, 48 + 1 + 6 + 3 + 6 = 82,
6942 = 481636, 48 + 16 + 36 = 102.
6942 = 481636 appears in the decimal expression of e:
e = 2.71828•••481636••• (from the 104033rd digit).
by Yoshio Mimura, Kobe, Japan
695
The smallest squares containing k 695's :
695556 = 8342,
141769569529 = 3765232,
695695169569156 = 263760342.
(406 / 695)2 = 0.341257698... (Komachic).
6952 = 483025, a square with different digits.
6953 = 335702375, and 32 + 32 + 52 + 72 + 02 + 232 + 72 +52 = 695.
(12 + 22 + ... + 362)(372 + 382 + ... + 1112)(1122 + 1132 + ... + 6952) = 8982985802.
3-by-3 magic squares consisting of different squares with constant 6952:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(3, 246, 650, 430, 510, 195, 546, 403, 150), | (6, 83, 690, 333, 606, 70, 610, 330, 45), |
(6, 342, 605, 435, 470, 270, 542, 381, 210), | (29, 78, 690, 150, 675, 70, 678, 146, 45), |
(38, 195, 666, 291, 610, 162, 630, 270, 115), | (45, 126, 682, 330, 605, 90, 610, 318, 99), |
(45, 290, 630, 358, 531, 270, 594, 342, 115), | (45, 290, 630, 470, 477, 186, 510, 414, 227), |
(45, 330, 610, 470, 435, 270, 510, 430, 195), | (66, 237, 650, 270, 610, 195, 637, 234, 150), |
(70, 285, 630, 378, 546, 205, 579, 322, 210), | (70, 285, 630, 450, 462, 259, 525, 434, 138), |
(90, 286, 627, 330, 573, 214, 605, 270, 210), | (150, 349, 582, 430, 510, 195, 525, 318, 326), |
(150, 430, 525, 461, 330, 402, 498, 435, 214) |
6952 = 483025, 4 + 8 + 30 + 2 + 5 = 72,
6952 = 483025, 4 + 830 + 2 + 5 = 292.
by Yoshio Mimura, Kobe, Japan
696
The smallest squares containing k 696's :
55696 = 2362,
696696025 = 263952,
69684696325696 = 83477362.
The squares which begin with 696 and end in 696 are
696784189696 = 8347362, 696830935696 = 8347642, 6960289191696 = 26382362,
6960436933696 = 26382642, 6962927677696 = 26387362,...
6962 = 484416, with 484 = 222, 4 = 22 and 16 = 42.
6962 + 6972 = 9852.
Komachi equations:
6962 = 122 / 32 * 42 * 52 * 62 + 72 * 82 * 92 = 92 * 872 / 62 / 52 * 42 / 32 * 22 * 102.
6962 = 363 + 563 + 643.
(13 + 23 + ... + 63)(73 + 83 + ... + 293)(303 + 313 + ... + 6963) = 22131620282,
(13 + 23 + ... + 113)(123 + 133 + ... + 2643)(2653 + 2663 + ... + 6963) = 5541294084482.
the square root of 696 is 26. 3 8 1 8 1 19 1 6 5 4 5 8 3 ...,
and 262 = 32 + 82 + 12 + 82 + 12 + 192 + 12 + 62 + 52 + 42 + 52 + 82 + 32.
6962 = 484416, 4 + 8 + 4 + 4 + 16 = 62,
6962 = 484416, 484 + 416 = 302.
by Yoshio Mimura, Kobe, Japan
697
The smallest squares containing k 697's :
169744 = 4122,
48697896976 = 2206762,
6973169736976 = 26406762.
(379 / 697)2 = 0.295673814... (Komachic).
6972 = 485809, a zigzag square.
6976 = 114655968874330129,
and 12 + 142 + 62 + 52 + 52 + 92 + 62 + 82 + 82 + 72 + 42 + 32 + 32 + 02 + 12 + 22 + 92 = 697.
(22 + 1)(52 + 1)(62 + 1)(102 + 1) = 6972 + 1.
9852 = 6962 + 6972.
(13 + 23 + ... + 6793)(6803 + 6813 + ... + 6973) = 176961115802.
3-by-3 magic squares consisting of different squares with constant 6972:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(0, 153, 680, 328, 600, 135, 615, 320, 72), | (12, 324, 617, 456, 463, 252, 527, 408, 204), |
(23, 132, 684, 468, 504, 113, 516, 463, 72), | (36, 112, 687, 212, 657, 96, 663, 204, 68), |
(57, 176, 672, 432, 537, 104, 544, 408, 153), | (68, 204, 663, 303, 608, 156, 624, 273, 148), |
(68, 408, 561, 489, 428, 252, 492, 369, 328), | (105, 372, 580, 428, 495, 240, 540, 320, 303), |
(148, 399, 552, 471, 468, 212, 492, 328, 369) |
6972 = 503 + 563 + 573,
6972 = 24 + 44 + 134 + 264.
6972 = 485809 appears in the decimal expressions of π and e:
π = 3.14159•••485809••• (from the 49117th digit),
e = 2.71828•••485809••• (from the 49994th digit).
by Yoshio Mimura, Kobe, Japan
698
The smallest squares containing k 698's :
698896 = 8362,
369869824 = 192322,
6989526986698881 = 836033912.
6982 = 383 + 473 + 693.
6982 = 487204, 4 + 8 + 7 + 2 + 0 + 4 = 52.
Komachi square sum : 6982 = 252 + 472+ 832 + 6912.
6982 + 6992 + 7002 + ... + 38762 = 1389412.
Page of Squares : First Upload August 18, 2005 ; Last Revised August 28, 2006by Yoshio Mimura, Kobe, Japan
699
The smallest squares containing k 699's :
376996 = 6142,
6991969924 = 836182,
1136996992216996 = 337193862.
6992 + 7002 + 7012 + ... + 33142 = 1096542.
6992 = 2332 + 4662 + 4662 : 6642 + 6642 + 3322 = 9962.
3-by-3 magic squares consisting of different squares with constant 6992:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(2, 166, 679, 406, 553, 134, 569, 394, 98), | (3, 144, 684, 396, 564, 117, 576, 387, 84), |
(7, 134, 686, 334, 602, 121, 614, 329, 58), | (10, 199, 670, 430, 530, 151, 551, 410, 130), |
(24, 261, 648, 432, 504, 219, 549, 408, 144), | (25, 310, 626, 374, 535, 250, 590, 326, 185), |
(26, 151, 682, 473, 506, 94, 514, 458, 121), | (46, 263, 646, 439, 514, 178, 542, 394, 199), |
(62, 119, 686, 154, 674, 103, 679, 142, 86), | (73, 326, 614, 374, 502, 311, 586, 361, 122), |
(74, 254, 647, 409, 542, 166, 562, 361, 206), | (86, 458, 521, 487, 334, 374, 494, 409, 278), |
(89, 206, 662, 298, 614, 151, 626, 263, 166), | (89, 322, 614, 406, 526, 217, 562, 329, 254), |
(94, 242, 649, 374, 569, 158, 583, 326, 206), | (108, 381, 576, 459, 396, 348, 516, 432, 189), |
(119, 362, 586, 454, 406, 343, 518, 439, 166) |
6992 = 488601, 4 + 8 + 8 + 60 + 1 = 92,
6992 = 488601, 488 + 601 = 332,
6992 = 488601, 48 + 8601 = 932.
by Yoshio Mimura, Kobe, Japan