370
The smallest squares containing k 370's :
370881 = 6092,
370370025 = 192452,
370370525370169 = 192450132.
3702 = 136900, where 1 = 12, 36 = 62, 900 = 302.
3702 = 136900, 1 + 369 + 0 + 0 = 1 + 369 + 0 * 0 = 370.
3702 = (12 + 1)(22 + 1)(1172 + 1)
= (12 + 1)(62 + 1)(432 + 1) = (32 + 1)(1172 + 1).
370k + 14430k + 27010k + 95090k are squares for k = 1,2,3 (3702, 999002, 297073002).
170k + 370k + 830k + 1130k are squares for k = 1,2,3 (502, 14602, 455002).
222k + 354k + 370k + 498k are squares for k = 1,2,3 (382, 7482, 151482).
(13 + 23 + 33 + ... + 1753)(1763 + 1773 + 1783 + ... + 3703) = 10300290002,
(13 + 23 + 33 + ... + 2943)(2953 + 2963 + 2973 + ... + 3703) = 23070180002.
by Yoshio Mimura, Kobe, Japan
371
The smallest squares containing k 371's :
23716 = 1542,
371371441 = 192712,
371371633710025 = 192710052.
3712 = 137641, 1 * 376 - 4 - 1 = 371.
12985k + 21518k + 46004k + 57134k are squares for k = 1,2,3 (3712, 775392, 172051252).
Komachi Fraction : 720 / 6193845 = (4 / 371)2.
3-by-3 magic squares consisting of different squares with constant 3712:
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3712 = 137641, 1 + 3 + 76 + 41 = 112,
3712 = 137641, 1 + 37 + 6 + 4 + 1 = 72.
3712 = 137641 appears in the decimal expression of π:
π = 3.14159•••137641••• (from the 44442nd digit).
by Yoshio Mimura, Kobe, Japan
372
The smallest squares containing k 372's :
3721 = 612,
13727637225 = 1171652,
370372372893721 = 192450612.
3722 = 138384, 1 + 383 - 8 - 4 = 138 / 3 * 8 + 4 = 372.
46k + 170k + 202k + 258k are squares for k = 1,2,3 (262, 3722, 55162).
3722 = 138384, 1 + 38 + 38 + 4 = 92,
3722 = 138384, 13 + 8 + 3 + 8 + 4 = 62,
3722 = 138384, 138 + 3 + 84 = 152.
3722 = 233 + 253 + 483.
Page of Squares : First Upload January 11, 2005 ; Last Revised March 2, 2011by Yoshio Mimura, Kobe, Japan
373
The smallest squares containing k 373's :
373321 = 6112,
13733730481 = 1171912,
3637333733537344 = 603103122.
The square root of 373 is 19.313..., 19 = 32 + 12 + 32.
3732 = 139129, 1 * 391 - 2 * 9 = 373.
3-by-3 magic squares consisting of different squares with constant 3732:
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3732 = 139129, 1 + 3 + 9 + 1 + 2 + 9 = 52,
3732 = 139129, 1 + 39 + 129 = 132,
3732 = 139129, 139 + 1 + 29 = 132.
by Yoshio Mimura, Kobe, Japan
374
The smallest squares containing k 374's :
374544 = 6122,
37476313744 = 1935882,
374773746137476 = 193590742.
3742 = 139876, a square with different digits.
11781k + 20757k + 32725k + 74613k are squares for k = 1,2,3 (3742, 848982, 214709662).
Page of Squares : First Upload January 11, 2005 ; Last Revised March 2, 2011by Yoshio Mimura, Kobe, Japan
375
The smallest squares containing k 375's :
337561 = 5812,
37529375625 = 1937252,
3753757163375569 = 612679132.
3752 = 140625, a zigzag square with different digits.
3752 = 253 + 503.
Komachi Square Sum : 3752 = 162 + 842 + 972 + 3522.
3-by-3 magic squares consisting of different squares with constant 3752:
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3752 = 140625, 1 + 4 + 0 + 6 + 25 = 62,
3752 = 140625, 14 + 0 + 62 + 5 = 92.
(1 + 2)(3 + 4 + 5 + ... + 12)(13 + 14 + 15 + ... + 37) = 3752.
Page of Squares : First Upload January 11, 2005 ; Last Revised February 16, 2009by Yoshio Mimura, Kobe, Japan
376
The smallest squares containing k 376's :
15376 = 1242,
337677376 = 183762,
37691937669376 = 61393762.
The squares which begin with 376 and end in 376 are
3767013376 = 613762, 37684127376 = 1941242, 376230117376 = 6133762,
376534413376 = 6136242, 376843743376 = 6138762,...
3767 = 1062465690325221376 :
12 + 02 + 62 + 22 + 42 + 62 + 52 + 62 + 92 + 02 + 32 + 22 + 52 + 22 + 22 + 12 + 32 + 72 + 62.
10998k + 22278k + 42018k + 66082k are squares for k = 1,2,3 (3762, 821562, 193685122).
Komachi Square Sum : 3762 = 22 + 42 + 52 + 92 + 812 + 3672.
3762 = 141376, 1 + 4 + 1 + 37 + 6 = 72,
3762 = 141376, 1 + 41 + 3 + 76 = 112.
3532 + 3542 + 3552 + 3562 + ... + 3762 = 17862.
Page of Squares : First Upload January 11, 2005 ; Last Revised March 2, 2011by Yoshio Mimura, Kobe, Japan
377
The smallest squares containing k 377's :
853776 = 9242,
1537737796 = 392142,
143772737737729 = 119905272.
3772 = 1562 + 2332 + 2522 : 2522 + 3322 + 6512 = 7732.
17k + 91k + 299k + 377k are squares for k = 1,2,3 (282, 4902, 90042).
Komachi Fraction : 3772 = 8954127 / 63.
3-by-3 magic squares consisting of different squares with constant 3772:
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3772 = 142129, 1 + 42 + 12 + 9 = 82,
3772 = 142129, 14 + 21 + 29 = 82.
3772 + 3782 + 3792 + 3802 + 3812 + ... + 10742 = 198932,
3772 + 3782 + 3792 + 3802 + 3812 + ... + 1180252 = 234100932.
3772 = 142129 appears in the decimal expression of π and e:
e = 2.71828•••142129••• (from the 26014th digit).
by Yoshio Mimura, Kobe, Japan
378
The smallest squares containing k 378's :
378225 = 6152,
61737837841 = 2484712,
3781858378137856 = 614968162.
3782 is the fourth square which is the sum of 7 sixth powers : (3,3,3,3,6,6,6).
3782 = (12 + 5)(22 + 5)(42 + 5)(112 + 5) = (22 + 5)(32 + 5)(42 + 5)(72 + 5)
= (22 + 5)(72 + 5)(172 + 5) = (32 + 5)(1012 + 5) = (42 + 5)(72 + 5)(112 + 5).
60k + 241k + 282k + 378k are squares for k = 1,2,3 (312, 5332, 95212).
3782 = 142884, 1 + 428 + 8 + 4 = 212,
3782 = 142884, 14 + 2 + 8 + 8 + 4 = 62,
3782 = 142884, 14 + 2 + 884 = 302.
3782 + 3792 + 3802 + 3812 + 3822 + ... + 63552 = 2924952.
(1 + 2)(3 + 4 + 5)(6 + 7 + 8)(9)(10 + 11) = 3782,
(1 + 2 + 3)(4 + 5)(6)(7)(8 + 9 + 10 + 11 + 12 + 13) = 3782,
(1)(2)(3)(4 + 5)(6)(7)(8 + 9 + 10 + 11 + 12 + 13) = 3782,
(1 + 2 + 3 + 4 + 5 + 6)(7)(8 + 9 + 10)(11 + 12 + 13) = 3782,
(1)(2 + 3 + 4)(5 + 6 + 7)(8 + 9 + 10 + 11 + 12 + 13)(14) = 3782,
(1 + 2)(3)(4 + 5)(6 + 7 + 8)(9 + 10 + 11 + 12 + 13 + 14 + 15) = 3782,
(1 + 2)(3 + 4 + 5 + 6 + 7 + 8 + 9)(10 + 11)(12 + 13 + 14 + 15) = 3782,
(1 + 2 + 3)(4 + 5)(6 + 7 + 8 + 9 + 10 + 11 + 12)(13 + 14 + 15) = 3782,
(1)(2)(3)(4 + 5)(6 + 7 + 8 + 9 + 10 + 11 + 12)(13 + 14 + 15) = 3782,
(1)(2 + 3 + 4 + ... + 22)(23 + 24 + 25 + ... + 40) = 3782,
(1)(2 + 3 + 4 + ... + 19)(20 + 21 + 22 + ... + 43) = 3782,
(1)(2 + 3 + 4 + ... + 10)(11 + 12 + 13 + ... + 73) = 3782,
(1 + 2 + 3 + ... + 8)(9 + 10 + 11 + ... + 89) = 3782.
by Yoshio Mimura, Kobe, Japan
379
The smallest squares containing k 379's :
379456 = 6162,
13793796 = 37142,
37985379379984 = 61632282.
3792 = 143641, 14 + 364 + 1 = 379.
3792 = 1142 + 2312 + 2782 : 8722 + 1322 + 4112 = 9732,
3792 = 1542 + 2312 + 2582 : 8522 + 1322 + 4512 = 9732.
3792 = 143641, 1 = 12, 4 = 22, 36 = 62.
Komachi Square Sum : 3792 = 22 + 62 + 192 + 582 + 3742.
3-by-3 magic squares consisting of different squares with constant 3792:
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3792 = 143641, 14 + 3 + 6 + 41 = 82,
3792 = 143641, 143 + 641 = 282.
3792 + 3802 + 3812 + 3822 + 3832 + ... + 4282 = 28552,
3792 + 3802 + 3812 + 3822 + 3832 + ... + 15222 = 340342,
3792 + 3802 + 3812 + 3822 + 3832 + ... + 315782 = 32398602.
3792 = 143641 appears in the decimal expression of π:
π = 3.14159•••143641••• (from the 11879th digit).
(143641 is the sixth 6-digit square in the expression of π.)
by Yoshio Mimura, Kobe, Japan