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220 - 229

220

The smallest squares containing k 220's :
2209 = 472,
220552201 = 148512,
362209220922025 = 190317952.

220 = (12 + 22 + 32 + ... + 872) / (12 + 22 + 32 + ... + 142).

(87 / 220)2 = 0.156384297... (Komachic).

2202 is the tenth square which is the sum of 9 sixth powers.

2695k + 10615k + 12485k + 22605k are squares for k = 1,2,3 (2202, 280502, 38357002).
4785k + 12265k + 14355k + 16995k are squares for k = 1,2,3 (2202, 258502, 31339002).

Komachi equation: 2202 = - 1 + 2 * 3456 * 7 + 8 + 9.

10222 = 1972 + 1982 + 1992 + 2002 + ... + 2202.

(1)(2 + 3)(4 + 5 + 6 + 7)(8)(9 + 10 + 11 + 12 + 13) = 2202,
(1)(2 + 3 + ... + 217)(218 + 219 + 220) = 39422,
(1 + 2 + ... + 10)(11 + 12 + ... + 220) = 11552,
(1 + 2 + ... + 72)(73)(74 + 75 + ... + 220) = 643862,
(1 + 2 + ... + 72)(73 + 74 + ... + 217)(218 + 219 + 220) = 1905302,
(1 + 2 + ... + 111)(112)(113 + 114 + ... + 220) = 1118882,
(1 + 2 + ... + 143)(144 + 145 + ... + 220) = 120122.

(13 + 23 + ... + 1433)(1443 + 1453 + ... + 2203) = 2267385122,
(13 + 23 + ... + 1753)(1763 + 1773 + ... + 2203) = 2896740002.

2202 = 48400 appears in the decimal expression of π:
  π = 3.14159•••48400••• (from the 74502nd digit).

Page of Squares : First Upload September 6, 2004 ; Last Revised February 22, 2011
by Yoshio Mimura, Kobe, Japan

221

The smallest squares containing k 221's :
221841 = 4712,
2210222169 = 470132,
22105221221161 = 47016192.

(201 / 221)2 = 0.827194365... (Komachic).

2213 - 2203 + 2193 - 2182 + .. + 13 = 23312.

2212 = 48841, a square with 3 kinds of digits.

2212 = 48841, a reversible square (14884 = 1222).

4420k + 7072k + 11713k + 25636k are squares for k = 1,2,3 (2212, 293932, 43468492).

2212 = 48841, 4 + 8 + 8 + 4 + 1 = 52,
2212 = 48841, 488 + 41 = 232.

2212 = 48841, an exchangeable square (14884 = 1222).

3-by-3 magic squares consisting of different squares with constant 2212:

028522042
10421802752
1952962402
      
36211721842
13921562722
16821042992

(1 + 2 + ... + 18)(19 + 20 + ... + 126)(127 + 128 + ... + 221) = 1487702,
(1 + 2 + ... + 60)(61 + 62 + ... + 83)(84 + 85 + ... + 221) = 2525402,
(1 + 2 + ... + 74)(75 + 76 + ... + 221) = 77702.

2212 = 48841 appears in the decimal expressions of π and e:
  π = 3.14159•••48841••• (from the 6401st digit),
  e = 2.71828•••48841••• (from the 228th digit),
  (48841 is the second 5-digit square in the expression of e.)

Page of Squares : First Upload September 6, 2004 ; Last Revised February 22, 2011
by Yoshio Mimura, Kobe, Japan

222

The smallest squares containing k 222's :
22201 = 1492,
12229042225 = 1105852,
122222200822225 = 110554152.

2222 is the 2nd square which is the sum of 8 seventh powers : (1, 1, 1, 1, 2, 4, 4, 4).

2222 = 49284, a zigzag square.

222k + 354k + 370k + 498k are squares for k = 1,2,3 (382, 7482, 151482).
154k + 222k + 258k + 810k are squares for k = 1,2,3 (382, 8922, 237322).
57k + 102k + 222k + 348k are squares for k = 1,2,3 (272, 4292, 73712).
1517k + 2257k + 12173k + 33337k are squares for k = 1,2,3 (2222, 355942, 62344262).

Komachi equations:
2222 = 9876 * 5 - 43 * 2 - 10,
2222 = 122 - 32 * 42 + 52 * 62 * 72 + 82 * 92 = 122 / 32 - 42 + 52 * 62 * 72 + 82 * 92
  = 122 / 32 / 42 * 52 * 62 * 72 + 82 * 92 = 122 / 32 * 452 / 62 * 72 + 82 * 92
  = - 122 + 32 * 42 + 52 * 62 * 72 + 82 * 92 = - 122 / 32 + 42 + 52 * 62 * 72 + 82 * 92.

2222 = 49284, 49 + 28 + 4 = 92,
2222 = 49284, 492 + 84 = 242.

(1 + 2 + ... + 12)(13 + 14 + ... + 92)(93 + 94 + ... + 222) = 819002,
(1 + 2 + ... + 27)(28 + 29 + ... + 152)(153 + 154 + ... + 222) = 2362502,
(1 + 2 + ... + 30)(31 + 32 + ... + 129)(130 + 131 + ... + 222) = 2455202,
(1 + 2 + ... + 49)(50 + 51 + ... + 97)(98 + 99 + ... + 222) = 2940002,
(1 + 2 + ... + 49)(50 + 51 + ... + 137)(138 + 139 + ... + 222) = 3927002,
(1 + 2 + ... + 100)(101)(102 + 103 + ... + 222) = 999902,
(1 + 2 + ... + 136)(137)(138 + 139 + ... + 222) = 1397402.

2222 = 49284 appears in the decimal expression of e:
  e = 2.71828•••49284••• (from the 81019th digit).

Page of Squares : First Upload September 6, 2004 ; Last Revised February 22, 2011
by Yoshio Mimura, Kobe, Japan

223

The smallest squares containing k 223's :
223729 = 4732,
122392223716 = 3498462,
223572232238224 = 149523322.

2232 = 49729, 4 * 9 * 7 - 29 = 223.

2232 = 49729, 4 + 9 + 7 + 29 = 72.

2232 = 49729, 49 = 72 and 729 = 272.

2232 = 49729 (4, 9 and 729 are squares).

2232 + 2242 + 2252 + 2262 + ... + 14702 = 325002.

3-by-3 magic square consisting of different squares with constant 2232:

 226622132
12321782542
18621172382

(1 + 2 + ... + 5)(6 + 7 + ... + 114)(115 + 116 + ... + 223) = 425102,
(1 + 2 + ... + 6)(7 + 8 + ... + 55)(56 + 57 + ... + 223) = 273422,
(1 + 2 + ... + 10)(11 + 12 + ... + 106)(107 + 108 + ... + 223) = 772202,
(1 + 2 + ... + 14)(15 + 16 + ... + 70)(71 + 72 + ... + 223) = 749702,
(1 + 2 + ... + 24)(25 + 26 + ... + 174)(175 + 176 + ... + 223) = 2089502,
(1 + 2 + ... + 29)(30 + 31 + ... + 95)(96 + 97 + ... + 223) = 1914002,
(1 + 2 + ... + 36)(37 + 38 + ... + 184)(185 + 186 + ... + 223) = 2943722,
(1 + 2 + ... + 37)(38 + 39 + ... + 109)(110 + 111 + ... + 223) = 2657342,
(1 + 2 + ... + 52)(53 + 54 + ... + 64)(65 + 66 + ... + 223) = 1488242,
(1 + 2 + ... + 55)(56 + 57 + ... + 91)(92 + 93 + ... + 223) = 2910602,
(1 + 2 + ... + 80)(81 + 82 + ... + 208)(209 + 210 + ... + 223) = 4406402,
(1 + 2 + ... + 98)(99 + 100 + ... + 154)(155 + 156 + ... + 223) = 6694382,
(1 + 2 + ... + 117)(118 + 119 + ... + 130)(131 + 132 + ... + 223) = 4279862,
(1 + 2 + ... + 136)(137 + 138 + ... + 187)(188 + 189 + ... + 223) = 7545962,
(1 + 2 + ... + 143)(144 + 145 + ... + 208)(209 + 210 + ... + 223) = 6177602,
(1 + 2 + ... + 216)(217)(218 + 219 + ... + 223) = 820262.

(13 + 23 + ... + 83)(93 + 103 + ... + 1123)(1133 + 1143 + ... + 2233) = 55039944962.

2232 = 49729 appears in the decimal expression of π:
  π = 3.14159•••49729••• (from the 10509th digit).

Page of Squares : First Upload September 6, 2004 ; Last Revised January 13, 2009
by Yoshio Mimura, Kobe, Japan

224

The smallest squares containing k 224's :
28224 = 1682,
2224820224 = 471682,
22242240604224 = 47161682.

The squares which begin with 224 and end in 224 are
2240318224 = 473322,   22400510224 = 1496682,   22449628224 = 1498322,
224043182224 = 4733322,   224361374224 = 4736682,...

2242 = 50176, a square with different digits.

2242 = 323 + 45 + 47.

2242 = 50176, 50 + 1 + 7 + 6 = 82.

2242 = 223 + 263 + 283.

A + B, A + C, A + D, B + C, B + D and C + D are squares for (A, B, C, D) = (224,1712,2912,7697).

Komachi equations:
2242 = 12 * 3 * 4 * 56 * 7 * 8 / 9,
2242 = 93 + 873 + 63 * 53 - 433 * 23 */ 13.

(1 + 2 + ... + 17)(18 + 19 + ... + 136)(137 + 138 + ... + 224) = 1492262,
(1 + 2 + ... + 49)(50 + 51 + ... + 55)(56 + 57 + ... + 224) = 955502,
(1 + 2 + ... + 50)(51 + 52 + ... + 149)(150 + 151 + ... + 224) = 4207502,
(1 + 2 + ... + 122)(123 + 124 + ... + 183)(184 + 185 + ... + 224) = 7653062.

2242 = 50176 appears in the decimal expression of π:
  π = 3.14159•••50176••• (from the 78259th digit).

Page of Squares : First Upload September 6, 2004 ; Last Revised January 6, 2011
by Yoshio Mimura, Kobe, Japan

225

The square of 15.

The smallest squares containing k 225's :
225 = 152,
126225225 = 112352,
1022596225225 = 10112352.

The squares which begin with 225 and end in 225 are
225450225 = 150152,   2250079225 = 474352,   2252926225 = 474652,
2254825225 = 474852,   2257675225 = 475152,...

2252 = 50625, 5 + 0 + 6 + 25 = 62,
2252 = 50625, 50 + 6 + 25 = 92.

2252 = 203 + 253 + 303

Komachi equations:
2252 = 94 * 84 * 74 / 64 * 54 / 44 * 34 / 214 = 94 * 84 / 74 / 64 * 54 / 44 / 34 * 214
  = 94 / 84 * 74 * 64 * 54 * 44 / 34 / 214 = 984 / 74 * 64 * 54 / 44 * 34 / 214.

225 is the sum of m squares for m = 1, 2,..., 211.

(1 + 2)(3 + 4 + 5 + 6 + 7)(8 + 9 + 10 + 11 + ... + 37) = 2252.

2252 + 2262 + 2272 + ... + 2982 = 22572,
2252 + 2262 + 2272 + ... + 3122 = 25302,
2252 + 2262 + 2272 + ... + 6312 = 89542,
2252 + 2262 + 2272 + ... + 704492 = 107958352.

3-by-3 magic squares consisting of different squares with constant 2252:

424722202
10321962402
20021002252
      
4212821852
145214021002
17221212802
      
25210022002
15221402892
16421452522
      
4029522002
11221842652
1912882802

(1 + 2)(3 + 4 + ... + 78)(79 + 80 + ... + 225) = 143642,
(1 + 2 + 3 + 4)(5 + 6 + ... + 199)(200 + 201 + ... + 225) = 331502,
(1 + 2 + ... + 9)(10 + 11 + ... + 150)(151 + 152 + ... + 225) = 846002,
(1 + 2 + ... + 15)(16 + 17 + ... + 159)(160 + 161 + ... + 225) = 1386002,
(1 + 2 + ... + 17)(18 + 19 + ... + 38)(39 + 40 + ... + 225) = 471242,
(1 + 2 + ... + 17)(18 + 19 + ... + 102)(103 + 104 + ... + 225) = 1254602,
(1 + 2 + ... + 26)(27 + 28 + ... + 117)(118 + 119 + ... + 225) = 2063882,
(1 + 2 + ... + 27)(28 + 29 + ... + 71)(72 + 73 + ... + 225) = 1372142,
(1 + 2 + ... + 42)(43 + 44 + ... + 161)(162 + 163 + ... + 225) = 3684242,
(1 + 2 + ... + 49)(50 + 51 + ... + 214)(215 + 216 + ... + 225) = 2541002.

(13 + 23 + ... + 1433)(1443 + 1453 + ... + 2253) = 2393511122.

2252 = 50625 appears in the decimal expression of π:
  π = 3.14159•••50625••• (from the 79586th digit).

Page of Squares : First Upload September 13, 2004 ; Last Revised May 14, 2010
by Yoshio Mimura, Kobe, Japan

226

The smallest squares containing k 226's :
226576 = 4762,
22622265649 = 1504072,
122697226226884 = 110768782.

2262 = 51076, a square with different digits.

2262 = 51076, 51 + 0 + 7 + 6 = 82.

2262 = 173 + 193 + 343.

(12 + 22 + 32 + ... + 1262) + (12 + 22 + 32 + ... + 2122) = (12 + 22 + 32 + ... + 2262).

226k + 610k + 822k + 1478k are squares for k = 1,2,3 (562, 18122, 634242).

Komachi equation: 2262 = 1 - 23 - 4 + 5678 * 9.

(1 + 2 + 3)(4 + 5 + ... + 111)(112 + 113 + ... + 226) = 269102,
(1 + 2 + ... + 4)(5 + 6 + ... + 16)(17 + 18 + ... + 226) = 56702,
(1 + 2 + ... + 28)(29 + 30 + ... + 34)(35 + 36 + ... + 226) = 438482,
(1 + 2 + ... + 29)(30 + 31 + ... + 34)(35 + 36 + ... + 226) = 417602,
(1 + 2 + ... + 39)(40 + 41 + ... + 96)(97 + 98 + ... + 226) = 2519402,
(1 + 2 + ... + 46)(47 + 48 + ... + 187)(188 + 189 + ... + 226) = 3794312,
(1 + 2 + ... + 51)(52 + 53 + ... + 221)(222 + 223 + ... + 226) = 1856402,
(1 + 2 + ... + 67)(68 + 69 + ... + 175)(176 + 177 + ... + 226) = 5535542,
(1 + 2 + ... + 90)(91 + 92 + ... + 130)(131 + 132 + ... + 226) = 5569202,
(1 + 2 + ... + 104)(105 + 106 + ... + 200)(201 + 202 + ... + 226) = 6661202,
(1 + 2 + ... + 121)(122 + 123 + ... + 139)(140 + 141 + ... + 226) = 5253932,
(1 + 2 + ... + 123)(124 + 125 + ... + 185)(186 + 187 + ... + 226) = 7854782.

Page of Squares : First Upload September 13, 2004 ; Last Revised February 22, 2011
by Yoshio Mimura, Kobe, Japan

227

The smallest squares containing k 227's :
222784 = 4722,
22272279121 = 1492392,
2272278074622724 = 476684182.

The first integer which is the sum of a square and a prime in 8 ways :
  22 + 223, 42 + 211, 62 + 191, 82 + 163, 102 + 127, 122 + 83, 142 + 31, 152 + 2.

2272 = 51529, a zigzag square.

Komachi equations:
2272 = 9 * 87 * 65 + 4 + 3 * 210,
2272 = 12 + 2342 + 52 - 62 - 72 * 82 - 92.

2272 = 51529, 5 + 15 + 29 = 72,
2272 = 51529, 53 + 153 + 293 = 1672.

2272 + 2282 + 2292 + 2302 + ... + 2592 = 13972.

3-by-3 magic squares consisting of different squares with constant 2272:

2211121982
15921422782
16221382792
      
624722222
14221742332
17721382342

(1 + 2 + ... + 14)(15 + 16 + ... + 84)(85 + 86 + ... + 227) = 900902,
(1 + 2 + ... + 80)(81 + 82 + 83 + 84)(85 + 86 + ... + 227) = 1544402.

2272 = 51529 appears in the decimal expressions of π and e:
  π = 3.14159•••51529••• (from the 54720th digit),
  e = 2.71828•••51529••• (from the 92009th digit).

Page of Squares : First Upload September 13, 2004 ; Last Revised May 14, 2010
by Yoshio Mimura, Kobe, Japan

228

The smallest squares containing k 228's :
22801 = 1512,
922822884 = 303782,
22862283228304 = 47814522.

12 + 22 + 32 + 42 + 52 + ... + 2282 = 3976814, different digits.

2282 is the 10th square which is the sum of 2 cubes : 2282 = 113 + 373.

2282 = 51984, a square with different digits.

2282 = (32 + 3)(42 + 3)(152 + 3).

228k + 4104k + 5700k + 7657k are squares for k = 1,2,3 (1332, 103932, 8386032).

Komachi equations:
2282 = 9 + 8765 + 43210,
2282 = 92 * 82 - 72 + 62 * 52 + 432 + 2102.

2282 = 51984, 5 + 19 + 8 + 4 = 62,
2282 = 51984, 513 + 93 + 83 + 43 = 3662,
2282 = 51984, 51 + 9 + 84 = 122.

(22 - 2)(32 - 2)(72 - 2)(92 - 2) = (2282 - 2).

(1 + 2 + ... + 104)(105 + 106 + ... + 174)(175 + 176 + ... + 228) = 7616702.

2282 = 51984 appears in the decimal expressions of π and e:
  π = 3.14159•••51984••• (from the 76032nd digit),
  e = 2.71828•••51984••• (from the 107774th digit).

Page of Squares : First Upload September 13, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

229

The smallest squares containing k 229's :
229441 = 4792,
61229522916 = 2474462,
229229229141124 = 151403182.

2292 = 52441, 5 + 2 + 4 + 4 + 1 = 42,
2292 = 52441, 524 + 4 + 1 = 232.

3-by-3 magic square consisting of different squares with constant 2292:

327622162
10421922692
2042992322

(1 + 2 + ... + 7)(8 + 9 + ... + 13)(14 + 15 + ... + 229) = 68042,
(1 + 2 + ... + 7)(8 + 9 + ... + 103)(104 + 105 + ... + 229) = 559442,
(1 + 2 + ... + 7)(8 + 9 + ... + 127)(128 + 129 + ... + 229) = 642602,
(1 + 2 + ... + 8)(9 + 10 + ... + 127)(128 + 129 + ... + 229) = 728282,
(1 + 2 + ... + 10)(11 + 12 + ... + 64)(65 + 66 + ... + 229) = 519752,
(1 + 2 + ... + 10)(11 + 12 + ... + 220)(221 + 222 + ... + 229) = 519752,
(1 + 2 + ... + 19)(20 + 21 + ... + 169)(170 + 171 + ... + 229) = 1795502,
(1 + 2 + ... + 21)(22 + 23 + ... + 42)(43 + 44 + ... + 229) = 628322,
(1 + 2 + ... + 35)(36 + 37 + ... + 70)(71 + 72 + ... + 229) = 1669502,
(1 + 2 + ... + 35)(36 + 37 + ... + 85)(86 + 87 + ... + 229) = 2079002,
(1 + 2 + ... + 38)(39 + 40 + ... + 95)(96 + 97 + ... + 229) = 2482352,
(1 + 2 + ... + 66)(67)(68 + 69 + ... + 229) = 596972,
(1 + 2 + ... + 126)(127)(128 + 129 + ... + 229) = 1360172,
(1 + 2 + ... + 143)(144 + 145 + ... + 220)(221 + 222 + ... + 229) = 5405402.

(13 + 23 + ... + 103)(113 + 123 + ... + 223)(233 + 243 + ... + 2293) = 3576711602.

2292 = 52441 appears in the decimal expression of e:
  e = 2.71828•••52441••• (from the 53847th digit).

Page of Squares : First Upload September 13, 2004 ; Last Revised January 13, 2009
by Yoshio Mimura, Kobe, Japan