Squares:380s

380

The smallest squares containing k 380's :
38025 = 195²,
3803805625 = 61675²,
38071380380401 = 6170201².

380² = 144400, with 144 = 12² and 400 = 20².

16815^k + 19285^k + 53295^k + 55005^k are squares for k = 1,2,3 (380², 80750², 18158300²).

Komachi Square Sum : 380² = 1² + 5² + 42² + 89² + 367².

380² + 381² + 382² + ... + 9979² = 575560².

Page of Squares : First Upload January 17, 2005 ; Last Revised March 4, 2011
by Yoshio Mimura, Kobe, Japan

381

The smallest squares containing k 381's :
238144 = 488²,
381381841 = 19529²,
381933812938129 = 19543127².

8890^k + 12446^k + 23368^k + 100457^k are squares for k = 1,2,3 (381², 104267², 32080581²).

3-by-3 magic squares consisting of different squares with constant 381²:

8²	61²	376²
236²	296²	43²
299²	232²	44²

16²	163²	344²
188²	296²	149²
331²	176²	68²

19²	212²	316²
244²	236²	173²
292²	211²	124²

28²	109²	364²
211²	308²	76²
316²	196²	83²

35²	160²	344²
256²	245²	140²
280²	244²	85²

54²	159²	342²
198²	306²	111²
321²	162²	126²

61²	152²	344²
184²	316²	107²
328²	149²	124²

64²	203²	316²
259²	256²	112²
272²	196²	181²

381² = 145161, 14 + 5 + 1 + 61 = 9²,
381² = 145161, 14 + 5 + 16 + 1 = 6².

381² = 7³ + 23³ + 51³ = 8³ + 30³ + 49³.

Page of Squares : First Upload January 17, 2005 ; Last Revised March 4, 2011
by Yoshio Mimura, Kobe, Japan

382

The smallest squares containing k 382's :
53824 = 232²,
3823938244 = 61838²,
382113826343824 = 19547732².

382² = 145924, 1 + 45 * 9 - 24 = 382.

18145^k + 22729^k + 50233^k + 54817^k are squares for k = 1,2,3 (382², 79838², 17583842²).

382² = 145924, 1 + 4 + 5 + 9 + 2 + 4 = 5²,
382² = 145924, 1 + 4 + 5924 = 77²,
382² = 145924, 1 + 459 + 24 = 22².

382² = 145924 appears in the decimal expression of e:
e = 2.71828•••145924••• (from the 17906th digit)

Page of Squares : First Upload November 21, 2005 ; Last Revised March 4, 2011
by Yoshio Mimura, Kobe, Japan

383

The smallest squares containing k 383's :
138384 = 372²,
3833838724 = 61918²,
3830538338343649 = 61891343².

383² = 146689, a square with a non-decreasing digits.

3-by-3 magic squares consisting of different squares with constant 383²:

6²	122²	363²
258²	267²	94²
283²	246²	78²

14²	138²	357²
213²	294²	122²
318²	203²	66²

18²	246²	293²
267²	202²	186²
274²	213²	162²

50²	165²	342²
267²	230²	150²
270²	258²	85²

69²	158²	342²
202²	309²	102²
318²	162²	139²

15²	60²	380²
164²	345²	48²
348²	160²	39²

383² = 146689, 1 + 466 + 8 + 9 = 22²,
383² = 146689, 14 + 66 + 89 = 13²,
383² = 146689, 146 + 6 + 8 + 9 = 13².

(1³ + 2³ + ... + 98³)(99³ + 100³ + ... + 224³)(225³ + 226³ + ... + 383³) = 8287153692048².

Page of Squares : First Upload January 17, 2005 ; Last Revised February 23, 2009
by Yoshio Mimura, Kobe, Japan

384

The smallest squares containing k 384's :
3844 = 62²,
384238404 = 19602²,
38438412816384 = 6199872².

The squares which begin with 384 and end in 384 are
38466192384 = 196128², 384241296384 = 619872², 384558736384 = 620128²,
384861418384 = 620372², 3840141898384 = 1959628²,...

1 / 384 = 0.0026041666..., 2² + 6² + 0² + 4² + 16² + 6² + 6² = 384.

20² + 384 = 28², 20² - 384 = 4².

Komachi equation: 384² = 1² / 2² / 3² * 4² * 56² / 7² * 8² * 9².

384² is the 4th square which is the sum of 9 seventh powers.

384² = 147456, 14 + 7 + 4 + 5 + 6 = 6²,
384² = 147456, 14 + 7 + 4 + 56 = 9²,
384² = 147456, 14 + 74 + 56 = 12².

384² = 16³ + 32³ + 48³.

Page of Squares : First Upload January 17, 2005 ; Last Revised July 22, 2011
by Yoshio Mimura, Kobe, Japan

385

The smallest squares containing k 385's :
33856 = 184²,
38503858176 = 196224²,
38566385313856 = 6210184².

385 = 1² + 2² + ... + 10².

385² = (13² + 6)(29² + 6).

385² = 28² + 29² + 30² + 31² + ... + 77².

385² is the 5th square which is the sum of 7 sixth powers : 1, 2, 4, 4, 6, 6, 6.

385² + 386² + 387² + ... + 5222² = 5223² + 5224² + 5225² + ... + 6579².

385^k + 1232^k + 2156^k + 2156^k are squares for k = 1,2,3 (77², 3311², 148225²).

3-by-3 magic squares consisting of different squares with constant 385²:

15²	60²	380²
164²	345²	48²
348²	160²	39²

15²	180²	340²
268²	249²	120²
276²	232²	135²

16²	225²	312²
255²	240²	160²
288²	200²	159²

20²	135²	360²
207²	300²	124²
324²	200²	57²

25²	96²	372²
240²	295²	60²
300²	228²	79²

36²	177²	340²
200²	300²	135²
327²	164²	120²

60²	180²	335²
240²	281²	108²
295²	192²	156²

385² = 148225, 14 + 8 + 2 + 25 = 7²,
385² = 148225, 14 + 8 + 22 + 5 = 7²,
385² = 148225, 14 + 82 + 25 = 11²,
385² = 148225, 14 + 822 + 5 = 29².

(1³ + 2³ + ... + 15³)(16³ + 17³ + ... + 21³)(22³ + 23³ + ... + 385³) = 1759998240².

(1³ + 2³ + ... + 144³)(145³ + 146³ + ... + 193³)(194³ + 195³ + ... + 385³) = 11665951577280².

385² = 148225 appears in the decimal expression of π:
π = 3.14159•••148225••• (from the 95333rd digit).

Page of Squares : First Upload January 17, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

386

The smallest squares containing k 386's :
386884 = 622²,
38638656 = 6216²,
3861638603863056 = 62142084².

386² + 1 is a prime number.

386² = 144² + 242² + 264² : 462² + 242² + 441² = 683².

Page of Squares : First Upload July 10, 2006 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

387

The smallest squares containing k 387's :
338724 = 582²,
27538738704 = 165948²,
1387497387763876 = 37249126².

387² = 143² + 242² + 266² : 662² + 242² + 341² = 783².

387²± 2 are primes.

3-by-3 magic squares consisting of different squares with constant 387²:

1²	62²	382²
242²	298²	49²
302²	239²	38²

10²	113²	370²
238²	290²	95²
305²	230²	62²

14²	122²	367²
158²	337²	106²
353²	146²	62²

17²	146²	358²
214²	302²	113²
322²	193²	94²

21²	168²	348²
228²	276²	147²
312²	213²	84²

22²	218²	319²
241²	242²	182²
302²	209²	122²

34²	118²	367²
143²	346²	98²
358²	127²	74²

38²	154²	353²
223²	298²	106²
314²	193²	118²

38²	154²	353²
239²	298²	62²
302²	193²	146²

48²	228²	309²
267²	204²	192²
276²	237²	132²

94²	223²	302²
242²	274²	127²
287²	158²	206²

387² = 149769, 1 + 4 + 9 + 7 + 6 + 9 = 6²,
387² = 149769, 14³ + 9³ + 7³ + 6³ + 9³ = 69²,
387² = 149769, 149 + 7 + 69 = 15².

25026^k + 30444^k + 40377^k + 53922^k are squares for k = 1,2,3 (387², 78045², 16324821²).

387² = 149769 appears in the decimal expression of e:
e = 2.71828•••149769••• (from the 37464th digit)

Page of Squares : First Upload November 21, 2005 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan

388

The smallest squares containing k 388's :
38809 = 197²,
23883538849 = 154543²,
1388038893883236 = 37256394².

Komachi Fraction : 388² = 8129376 / 54.

Komachi equations:
388² = - 1⁴ + 2⁴ + 3⁴ + 4⁴ * 5⁴ - 6⁴ + 7⁴ - 8⁴ - 9⁴ = - 9⁴ - 8⁴ + 7⁴ - 6⁴ + 5⁴ * 4⁴ + 3⁴ + 2⁴ - 1⁴.

388² = 8⁴ + 12⁴ + 12⁴ + 18⁴.

388² = 150544, 1 + 5 + 0 + 54 + 4 = 8²,
388² = 150544, 1 + 50 + 5 + 4 + 4 = 8²,
388² = 150544, 1 + 50 + 5 + 44 = 10²,
388² = 150544, 15 + 0 + 5 + 44 = 8².

Page of Squares : First Upload January 17, 2005 ; Last Revised July 4, 2010
by Yoshio Mimura, Kobe, Japan

389

The smallest squares containing k 389's :
389376 = 624²,
3891389161 = 62381²,
389389157838916 = 19732946².

389² = 151321, 15 * 13 * 2 - 1 = 389.

389² = 99² + 264² + 268² : 862² + 462² + 99² = 983².

3-by-3 magic squares consisting of different squares with constant 389²:

12²	61²	384²
99²	372²	56²
376²	96²	27²

12²	204²	331²
236²	267²	156²
309²	196²	132²

20²	195²	336²
264²	240²	155²
285²	236²	120²

36²	93²	376²
168²	344²	69²
349²	156²	72²

389² = 151321, 15² + 1² + 3² + 21² = 26²,
389² = 151321, 15 + 1 + 32 + 1 = 7²,
389² = 151321, 15 + 13 + 21 = 7².

52² + 53² + 54² + 55² + ... + 389² = 4433².

(1² + 2² + ... + 6²)(7² + 8² + ... + 39²)(40² + 41² + ... + 389²) = 6051045².

(1³ + 2³ + ... + 143³)(144³ + 145³ + ... + 389³) = 773775288².

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