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370 - 379

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.

Page of Squares : First Upload January 11, 2005 ; Last Revised November 2, 2013
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:

2211123542
18623062972
32121782542
     
6217723262
241224621382
282221421112
     
6219123182
214225821592
303218621062
     
15211023542
25422552902
27022462652
33210623542
17423182792
32621592782
     
81222222862
254217422072
258224121142

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).

Page of Squares : First Upload January 11, 2005 ; Last Revised March 2, 2011
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, 2011
by 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:

 4216823332
237225621322
288221321042
     
36213223472
22822832842
293220421082
     
48214823392
171231221122
328214121082
     
48223722842
256222821472
267217621922

3732 = 139129, 1 + 3 + 9 + 1 + 2 + 9 = 52,
3732 = 139129, 1 + 39 + 129 = 132,
3732 = 139129, 139 + 1 + 29 = 132.

Page of Squares : First Upload January 11, 2005 ; Last Revised February 16, 2009
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, 2011
by 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:

0210523602
22522882842
30022162632
   
226123702
20523102502
31422022352
   
228623652
21123022702
31022052502
   
13213423502
170231021252
33421632502
   
1925823702
11023552502
35821062352
2628223652
11823492702
35521102502
   
35213023502
26222592702
266223821152
   
50212523502
19423102832
317217021062
   
50221823012
250220521902
275222621182
   
77219023142
230227521102
286217021732

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, 2009
by 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, 2011
by 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:

0214523482
260225221052
273224021002
   
12215623432
224227321322
30322082842
   
17214423482
192230321162
32421722872
   
24214323482
177231221162
33221562872
   
44220723122
228226421432
297217221562

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).

Page of Squares : First Upload January 11, 2005 ; Last Revised August 17, 2013
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.

Page of Squares : First Upload January 11, 2005 ; Last Revised December 7, 2013
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:

929823662
20623062872
31822012462
   
4228923662
11423542732
35921022662
   
46218623272
249226221142
282220121542
   
54219923182
233223421862
29422222892
   
66222622972
249219822062
278223121142

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.)

Page of Squares : First Upload January 11, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan