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690 - 699

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:

02 12 82252
112222 92 22
132142172 62
202 32162 52
     
02 12172202
42192132122
72182142112
252 22 62 52
     
12 32142222
42202152 72
122162132112
232 52102 62
     
12 62132222
82232 42 92
152 22192102
202112122 52
     
12 62132222
82232 42 92
152102192 22
202 52122112

6902 = 476100 appears in the decimal expression of π:
  π = 3.14159•••476100••• (from the 30316th digit).

Page of Squares : First Upload August 18, 2005 ; Last Revised December 7, 2013
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:

A2B2C2
D2E2F2
G2H2K2
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, 2011
by 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:

A2B2C2
D2E2F2
G2H2K2
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).

Page of Squares : First Upload August 18, 2005 ; Last Revised August 17, 2013
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).

Page of Squares : First Upload August 18, 2005 ; Last Revised January 16, 2014
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:

A2B2C2
D2E2F2
G2H2K2
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.

Page of Squares : First Upload August 18, 2005 ; Last Revised July 27, 2009
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.

Page of Squares : First Upload August 18, 2005 ; Last Revised June 25, 2010
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:

A2B2C2
D2E2F2
G2H2K2
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).

Page of Squares : First Upload August 18, 2005 ; Last Revised July 27, 2009
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, 2006
by 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:

A2B2C2
D2E2F2
G2H2K2
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.

Page of Squares : First Upload August 18, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan