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610 - 619

610

The smallest squares containing k 610's :
36100 = 1902,
161036100 = 126902,
206106106108816 = 143563962.

226k + 610k + 822k + 1478k are squares for k = 1,2,3 (562, 18122, 634242).
230k + 610k + 665k + 2720k are squares for k = 1,2,3 (652, 28752, 1437252).

(12 + 22 + ... + 4392) + (12 + 22 + ... + 5222) = (12 + 22 + ... + 6102).

Page of Squares : First Upload January 25, 2006 ; Last Revised March 17, 2011
by Yoshio Mimura, Kobe, Japan

611

The smallest squares containing k 611's :
206116 = 4542,
51611661124 = 2271822,
2576116110746116 = 507554542.

(12 + 3)(22 + 3)(82 + 3)(142 + 3) = 6112 + 3.

(13 + 23 + ... + 1883)(1893 + 1903 + ... + 6113) = 33066079202.

6112 = 720 + 721 + 723.

3-by-3 magic squares consisting of different squares with constant 6112:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(1, 78, 606, 354, 494, 63, 498, 351, 46),(1, 186, 582, 426, 417, 134, 438, 406, 129),
(6, 143, 594, 242, 546, 129, 561, 234, 62),(15, 330, 514, 370, 414, 255, 486, 305, 210),
(18, 154, 591, 246, 543, 134, 559, 234, 78),(26, 273, 546, 399, 406, 222, 462, 366, 161),
(66, 399, 458, 426, 298, 321, 433, 354, 246),(78, 234, 559, 321, 494, 162, 514, 273, 186),
(78, 351, 494, 426, 386, 207, 431, 318, 294),(81, 262, 546, 414, 426, 143, 442, 351, 234)

6112 = 373321, 32 + 72 + 32 + 32 + 22 + 12 = 92,
6112 = 373321, 3 + 7 + 33 + 21 = 82,
6112 = 373321, 37 + 3 + 3 + 21 = 82,
6112 = 373321, 3 + 73 + 3 + 21 = 102,
6112 = 373321, 37 + 3 + 321 = 192.

6112 = 373321 appears in the decimal expression of π:
  π = 3.14159•••373321••• (from the 62819th digit).

Page of Squares : First Upload June 20, 2005 ; Last Revised August 29, 2011
by Yoshio Mimura, Kobe, Japan

612

The smallest squares containing k 612's :
16129 = 1272,
612612001 = 247512,
61232612566129 = 78251272.

6122 = (12 + 8)(22 + 8)(32 + 8)(142 + 8) = (32 + 8)(102 + 8)(142 + 8).

Komachi equations
6122 = 12 * 22 * 342 * 562 / 72 / 82 * 92 = 12 * 22 * 342 / 562 * 72 * 82 * 92.

6122 = 64 + 124 + 124 + 244 = 44 + 84 + 144 + 244.

6122 + 6132 + 6142 + 6152 + ... + 6442 = 36082,
6122 + 6132 + 6142 + 6152 + ... + 19092 = 473772,
6122 + 6132 + 6142 + 6152 + ... + 22042 = 591182,
6122 + 6132 + 6142 + 6152 + ... + 35022 = 1193572,
6122 + 6132 + 6142 + 6152 + ... + 37902 = 1344532,
6122 + 6132 + 6142 + 6152 + ... + 46672 = 1838982,
6122 + 6132 + 6142 + 6152 + ... + 75252 = 3768132,
6122 + 6132 + 6142 + 6152 + ... + 95892 = 5420972,
6122 + 6132 + 6142 + 6152 + ... + 124112 = 7982702,
6122 + 6132 + 6142 + 6152 + ... + 1915802 = 484134822.

(1 + 2 + ... + 8)(9 + 10 + ... + 144) = 6122.

(13 + 23 + ... + 583)(593 + 603 + ... + 1173)(1183 + 1193 + ... + 6123) = 21449017636202.

Page of Squares : First Upload June 20, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

613

The smallest squares containing k 613's :
613089 = 7832,
344613613444 = 5870382,
61306613661316 = 78298542.

(415 / 613)2 = 0.458326791... (Komachic).

6132 = 375769, a zigzag square.

6133 - 6123 + 6113 - 6103 + ... + 13 = 107452.

6132 = 13 + 323 + 703, the 8th square which is the sum of three cubes.

6132 = 703 + 85 + 17.

3-by-3 magic squares consisting of different squares with constant 6132:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(12, 260, 555, 405, 420, 188, 460, 363, 180),(27, 208, 576, 336, 477, 188, 512, 324, 93),
(28, 216, 573, 384, 453, 152, 477, 352, 156),(64, 372, 483, 408, 387, 244, 453, 296, 288),
(72, 188, 579, 219, 552, 152, 568, 189, 132),(72, 197, 576, 323, 504, 132, 516, 288, 163),
(120, 315, 512, 387, 440, 180, 460, 288, 285) 

6132 = 375769, 37 + 5 + 7 + 6 + 9 = 82,
6132 = 375769, 3 + 7 + 5 + 76 + 9 = 102,
6132 = 375769, 3 + 75 + 7 + 6 + 9 = 102,
6132 = 375769, 3 + 7 + 5 + 769 = 282.

6132 = 375769 appears in the decimal expression of π:
  π = 3.14159•••375769••• (from the 61862nd digit).

Page of Squares : First Upload June 20, 2005 ; Last Revised June 29, 2009
by Yoshio Mimura, Kobe, Japan

614

The smallest squares containing k 614's :
614656 = 7842,
361456144 = 190122,
614661461462689 = 247923672.

6142 = 376996, 3 + 7 + 6 + 9 + 96 = 112,
6142 = 376996, 3 + 7 + 6 + 99 + 6 = 112,
6142 = 376996, 37 + 69 + 9 + 6 = 112,
6142 = 376996, 376 + 9 + 9 + 6 = 202.

Page of Squares : First Upload January 25, 2006 ; Last Revised January 6, 2011
by Yoshio Mimura, Kobe, Japan

615

The smallest squares containing k 615's :
61504 = 2482,
35615615841 = 1887212,
1096156615361536 = 331082562.

Komachi square sum : 6152 = 32 + 962 + 1742 + 5822.

(13 + 23 + ... + 603)(613 + 623 + ... + 1643)(1653 + 1663 + ... + 6153) = 46350532800002.

6152 = 43 + 173 + 723.

3-by-3 magic squares consisting of different squares with constant 6152:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(0, 135, 600, 369, 480, 108, 492, 360, 81),(10, 110, 605, 275, 542, 94, 550, 269, 58),
(10, 110, 605, 418, 445, 74, 451, 410, 82),(14, 77, 610, 173, 586, 70, 590, 170, 35),
(14, 125, 602, 355, 490, 110, 502, 350, 61),(19, 290, 542, 410, 445, 110, 458, 310, 269),
(35, 170, 590, 310, 515, 130, 530, 290, 115),(35, 218, 574, 310, 490, 205, 530, 301, 82),
(35, 310, 530, 370, 413, 266, 490, 334, 163),(35, 370, 490, 422, 371, 250, 446, 322, 275),
(38, 290, 541, 334, 445, 262, 515, 310, 130),(60, 207, 576, 360, 480, 135, 495, 324, 168),
(70, 338, 509, 365, 434, 238, 490, 275, 250),(74, 157, 590, 250, 550, 115, 557, 226, 130),
(82, 205, 574, 350, 490, 125, 499, 310, 182),(82, 326, 515, 410, 355, 290, 451, 382, 170),
(110, 269, 542, 355, 458, 206, 490, 310, 205),(115, 250, 550, 290, 514, 173, 530, 227, 214),
(115, 290, 530, 394, 445, 158, 458, 310, 269) 

6152 = 378225, 37 + 82 + 25 = 122.

Page of Squares : First Upload June 20, 2005 ; Last Revised June 29, 2009
by Yoshio Mimura, Kobe, Japan

616

The smallest squares containing k 616's :
41616 = 2042,
1616361616 = 402042,
291616961625616 = 170767962.

The squares which begin with 616 and end in 616 are
61605225616 = 2482042,   61650903616 = 2482962,   616545321616 = 7852042,
616689807616 = 7852962,   6161336697616 = 24822042,...

6162 = 379456, a square with different digits.

6162 = 379456, 3 + 7 + 945 + 6 = 312.

Page of Squares : First Upload June 20, 2005 ; Last Revised August 21, 2006
by Yoshio Mimura, Kobe, Japan

617

The smallest squares containing k 617's :
76176 = 2762,
61717961761 = 2484312,
1061761761718596 = 325846862.

(13 + 23 + ... + 1363)(1373 + 1383 + ... + 4113)(4123 + 4133 + ... + 6173) = 1339176566019602.

6172 = 380689, 3 + 80 + 6 * 89 = 617.

3-by-3 magic squares consisting of different squares with constant 6172:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(12, 231, 572, 319, 492, 192, 528, 292, 129),(24, 108, 607, 292, 537, 84, 543, 284, 72),
(33, 140, 600, 360, 492, 95, 500, 345, 108),(33, 184, 588, 248, 543, 156, 564, 228, 103),
(39, 228, 572, 348, 481, 168, 508, 312, 159),(60, 345, 508, 383, 420, 240, 480, 292, 255),
(63, 212, 576, 432, 396, 193, 436, 423, 108),(68, 336, 513, 423, 348, 284, 444, 383, 192),
(72, 311, 528, 392, 432, 201, 471, 312, 248) 

6172 = 380689 appears in the decimal expression of π:
  π = 3.14159•••380689••• (from the 82289th digit).

Page of Squares : First Upload June 20, 2005 ; Last Revised June 29, 2009
by Yoshio Mimura, Kobe, Japan

618

The smallest squares containing k 618's :
1361889 = 11672,
2618061889 = 511672,
618697618618689 = 248736332.

6182 = 381924, a zigzag square with different digits.

47586k + 79722k + 84666k + 169950k are squares for k = 1,2,3 (6182, 2113562, 782944202).

Komachi Square Sum : 6182 = 32 + 42 + 72 + 92 + 852 + 6122.

6182 = 332 + 523 + 593.

The 4-by-4 magic square consisting of different squares with constant 618:

02 12162192
52182132102
82172122112
232 22 72 62

6182 = 381924, 3 + 8 + 19 + 2 + 4 = 62,
6182 = 381924, 38 + 19 + 24 = 92.

Page of Squares : First Upload June 20, 2005 ; Last Revised March 17, 2011
by Yoshio Mimura, Kobe, Japan

619

The smallest squares containing k 619's :
236196 = 4862,
16198161984 = 1272722,
27636196196196 = 52570142.

240k + 306k + 313k + 366k are squares for k = 1,2,3 (352, 6192, 110532).

3-by-3 magic squares consisting of different squares with constant 6192:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(6, 282, 551, 313, 474, 246, 534, 281, 138),(7, 126, 606, 174, 582, 119, 594, 169, 42),
(9, 78, 614, 398, 471, 54, 474, 394, 57),(18, 231, 574, 434, 414, 153, 441, 398, 174),
(34, 246, 567, 378, 441, 214, 489, 358, 126),(57, 286, 546, 394, 471, 78, 474, 282, 281),
(114, 286, 537, 362, 471, 174, 489, 282, 254),(142, 279, 534, 306, 506, 183, 519, 222, 254)

6192 = 383161, 33 + 83 + 33 + 13 + 63 + 13 = 282,
6192 = 383161, 38 + 3 + 1 + 6 + 1 = 72,
6192 = 383161, 3 + 8 + 31 + 6 + 1 = 72,
6192 = 383161, 383 + 16 + 1 = 202,
6192 = 383161, 3 + 831 + 6 + 1 = 292.

Page of Squares : First Upload January 25, 2006 ; Last Revised March 17, 2011
by Yoshio Mimura, Kobe, Japan