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240 - 249

240

The smallest squares containing k 240's :
2401 = 492,
112402404 = 106022,
202402407312400 = 142268202.

2402 = 103 + 123 + 383.

2402 = (22 - 1)(32 - 1)(492 - 1) = (42 - 1)(72 - 1)(92 - 1) = (52 - 1)(492 - 1).

If (A, B, C, D) = (2402,2522,2752), then A, B, C, A + B, B + C and C + A are squares.

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

172 + 240 = 232, 172 - 240 = 72.

Komachi equations:
2402 = 9 * 8 * 765 + 4 * 3 * 210,
2402 = 92 * 82 * 72 / 62 * 52 * 42 * 32 / 212 = 92 * 82 / 72 / 62 * 52 * 42 / 32 * 212
2402 = 982 / 72 * 62 * 52 * 42 * 32 / 212.

(1 + 2 + ... + 44)(45 + 46 + ... + 55)(56 + 57 + ... + 240) = 1221002,
(1 + 2 + ... + 44)(45 + 46 + ... + 144)(145 + 146 + ... + 240) = 4158002,
(1 + 2 + ... + 48)(49 + 50 + ... + 240) = 57122,
(1 + 2 + ... + 54)(55 + 56 + ... + 209)(210 + 211 + ... + 240) = 4603502,
(1 + 2 + ... + 88)(89 + 90 + ... + 177)(178 + 179 + ... + 240) = 7812422,
(1 + 2 + ... + 120)(121 + 122 + ... + 240) = 125402.

(12 + 22 + ... + 1202)(1212 + 1222 + ... + 2402) = 15375802.

2402 = 57600 appears in the decimal expressions of π and e:
  π = 3.14159•••57600••• (from the 78919th digit),
  e = 2.71828•••57600••• (from the 31492nd digit).

Page of Squares : First Upload October 4, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

241

The smallest squares containing k 241's :
6241 = 792,
191241241 = 138292,
12410324163241 = 35228292.

The squares which begin with 241 and end in 241 are
2417787241 = 491712,   24127098241 = 1553292,   24155687241 = 1554212,
241003428241 = 4909212,   241158584241 = 4910792,...

241 is the fourth prime for which the Legendre Symbol (a/241) = 1 for a = 1, 2, 3, 4, 5, 6.

2412 is the 10th square which is the sum of 7 fifth powers : (1, 3, 3, 3, 6, 7, 8).

60k + 241k + 282k + 378k are squares for k = 1,2,3 (312, 5332 95212).
125k + 241k + 305k + 485k are squares for k = 1,2,3 (342, 6342, 125862).
210k + 241k + 538k + 692k are squares for k = 1,2,3 (412, 9332, 225912).

2412 = 58081, a zigzag square pegged by 8.

3-by-3 magic squares consisting of different squares with constant 2412:

924822362
12822012362
20421242332
      
16213222012
15621592922
18321242962

Komachi equations:
2412 = - 982 + 72 + 652 * 42 + 32 * 22 * 12 = - 982 + 72 + 652 * 42 + 32 * 22 / 12.

(1 + 2 + 3)(4 + 5 + ... + 115)(116 + 117 + ... + 241) = 299882,
(1 + 2 + ... + 16)(17 + 18 + ... + 34)(35 + 36 + ... + 241) = 422282,
(1 + 2 + ... + 16)(17 + 18 + ... + 115)(116 + 117 + ... + 241) = 1413722,
(1 + 2 + ... + 38)(39 + 40 + ... + 57)(58 + 59 + ... + 241) = 1363442,
(1 + 2 + ... + 45)(46 + 47 + ... + 206)(207 + 208 + ... + 241) = 4057202,
(1 + 2 + ... + 48)(49 + 50 + ... + 96)(97 + 98 + ... + 241) = 3166802,
(1 + 2 + ... + 80)(81 + 82 + ... + 126)(127 + 128 + ... + 241) = 5713202,
(1 + 2 + ... + 125)(126)(127 + 128 + ... + 241) = 1449002,
(1 + 2 + ... + 144)(145 + 146 + ... + 193)(194 + 195 + ... + 241) = 9500402.

(12 + 22 + ... + 1202)(1212 + 1222 + ... + 2402)(2412) = 3705567802.

(13 + 23 + ... + 73)(83 + 93 + 103)(113 + 123 + ... + 2413) = 386527682.

2412 = 58081 appears in the decimal expressions of π and e:
  π = 3.14159•••58081••• (from the 44434th digit),
  e = 2.71828•••58081••• (from the 73460th digit).

Page of Squares : First Upload October 4, 2004 ; Last Revised February 23, 2011
by Yoshio Mimura, Kobe, Japan

242

The smallest squares containing k 242's :
242064 = 4922,
24229724281 = 1556592,
24224292425124 = 49218182.

2422 = 58564, a zigzag square.

242, 243, 244 and 245 are four consecutive integers having square factors (the first case).

178k + 242k + 494k + 850k are squares for k = 1,2,3 (422, 10282, 274682).

Komachi equation: 2422 = - 13 * 23 * 343 - 53 + 63 - 73 + 83 * 93.

2422 = 58564, 5 + 85 + 6 + 4 = 102.

2422 = 43 + 253 + 353.

2422 = 114 + 114 + 114 + 114.

(1)(2 + 3 + ... + 62)(63 + 64 + ... + 242) = 73202,
(1 + 2 + ... + 4)(5 + 6 + ... + 80)(81 + 82 + ... + 242) = 290702,
(1 + 2 + ... + 4)(5 + 6 + ... + 157)(158 + 159 + ... + 242) = 459002,
(1 + 2 + ... + 11)(12 + 13 + ... + 132)(133 + 134 + ... + 242) = 1089002,
(1 + 2 + ... + 17)(18 + 19 + ... + 157)(158 + 159 + ... + 242) = 1785002,
(1 + 2 + ... + 61)(62)(63 + 64 + ... + 242) = 567302,
(1 + 2 + ... + 125)(126 + 127 + ... + 217)(218 + 219 + ... + 242) = 8452502,
(1 + 2 + ... + 144)(145)(146 + 147 + ... + 242) = 1687802,
(1 + 2 + ... + 203)(204 + 205 + ... + 221)(222 + 223 + ... + 242) = 6211802.

(12 + 22 + 32 + ... + 392) + (12 + 22 + 32 + ... + 482) = (12 + 22 + 32 + 42 + ... + 2422).

(13 + 23 + ... + 1323)(1333 + 1343 + ... + 1533)(1543 + 1553 + ... + 2423) = 18580872229922.

2422 = 58564 appears in the decimal expressions of π and e:
  π = 3.14159•••58564••• (from the 11218th digit),
  e = 2.71828•••58564••• (from the 16853rd digit).

Page of Squares : First Upload October 4, 2004 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

243

The smallest squares containing k 243's :
24336 = 1562,
12430243081 = 1114912,
2431216243243225 = 493073652.

2432 = (2 + 4 + 3)5.

2432 = 59049, 5 + 90 + 49 = 122.

2432 = 273 + 273 + 273.

2432 = 39 + 39 + 39 = 273 + 273 + 273.

Komachi equations:
2432 = 125 - 35 * 45 + 565 - 75 * 85 + 95 = 125 - 35 * 45 + 565 / 75 - 85 + 95
  = 125 - 35 * 45 + 565 / 75 / 85 * 95 = 125 - 35 * 45 - 565 + 75 * 85 + 95
  = 125 - 35 * 45 - 565 / 75 + 85 + 95 = 125 - 35 * 45 * 565 / 75 / 85 + 95
  = 125 - 35 * 45 / 565 * 75 * 85 + 95 = 125 / 35 - 45 + 565 - 75 * 85 + 95
  = 125 / 35 - 45 + 565 / 75 - 85 + 95 = 125 / 35 - 45 + 565 / 75 / 85 * 95
  = 125 / 35 - 45 - 565 + 75 * 85 + 95 = 125 / 35 - 45 - 565 / 75 + 85 + 95
  = 125 / 35 - 45 * 565 / 75 / 85 + 95 = 125 / 35 - 45 / 565 * 75 * 85 + 95
  = 125 / 35 / 45 - 565 / 75 / 85 + 95 = 125 / 35 / 45 * 565 - 75 * 85 + 95
  = 125 / 35 / 45 * 565 / 75 - 85 + 95 = 125 / 35 / 45 * 565 / 75 / 85 * 95
  = 125 / 35 / 45 / 565 * 75 * 85 * 95 = - 125 + 35 * 45 - 565 + 75 * 85 + 95
  = - 125 + 35 * 45 - 565 / 75 + 85 + 95 = - 125 + 35 * 45 + 565 - 75 * 85 + 95
  = - 125 + 35 * 45 + 565 / 75 - 85 + 95 = - 125 + 35 * 45 + 565 / 75 / 85 * 95
  = - 125 + 35 * 45 * 565 / 75 / 85 + 95 = - 125 + 35 * 45 / 565 * 75 * 85 + 95
  = - 125 / 35 + 45 - 565 + 75 * 85 + 95 = - 125 / 35 + 45 - 565 / 75 + 85 + 95
  = - 125 / 35 + 45 + 565 - 75 * 85 + 95 = - 125 / 35 + 45 + 565 / 75 - 85 + 95
  = - 125 / 35 + 45 + 565 / 75 / 85 * 95 = - 125 / 35 + 45 * 565 / 75 / 85 + 95
  = - 125 / 35 + 45 / 565 * 75 * 85 + 95 = - 125 / 35 / 45 + 565 / 75 / 85 + 95
  = - 125 / 35 / 45 * 565 + 75 * 85 + 95 = - 125 / 35 / 45 * 565 / 75 + 85 + 95.

3-by-3 magic squares consisting of different squares with constant 2432:

224922382
16121782382
18221582312
    
328422282
13221922692
20421232482
    
17210622182
13421782972
20221272462
    
2229422232
11322022742
2142972622
    
22212722062
146215821132
19321342622

(1 + 2 + ... + 10)(11 + 12 + ... + 45)(46 + 47 + ... + 243) = 392702,
(1 + 2 + ... + 23)(24 + 25 + ... + 45)(46 + 47 + ... + 243) = 774182,
(1 + 2 + ... + 32)(33 + 34 + ... + 45)(46 + 47 + ... + 243) = 875162,
(1 + 2 + ... + 32)(33 + 34 + ... + 197)(198 + 199 + ... + 243) = 3187802,
(1 + 2 + ... + 242)(243) = 26732.

2432 = 59049 appears in the decimal expression of e:
  e = 2.71828•••59049••• (from the 82903rd digit).

Page of Squares : First Upload October 4, 2004 ; Last Revised May 18, 2010
by Yoshio Mimura, Kobe, Japan

244

The smallest squares containing k 244's :
26244 = 1622,
522442449 = 228572,
244244948642244 = 156283382.

The squares which begin with 244 and end in 244 are
24441570244 = 1563382,   244196082244 = 4941622,   244370058244 = 4943382,
244690494244 = 4946622,   244864646244 = 4948382,...

2442 = 59536, 59 + 5 + 36 = 102.

2442 = 59536, a square with odd digits except the last digit 6.

2442 is the 5th square which is the sum of 4 fifth powers : 15 + 35 + 35 + 95.

2442 = 13 + 63 + 393 = 15 + 35 + 35 + 95.

Komachi Square Sums : 2442 = 12 + 52 + 62 + 72 + 482 + 2392
  = 12 + 52 + 72 + 92 + 682 + 2342 = 62 + 72 + 82 + 92 + 352 + 2412.

2442 + 2452 + 2462 + 2472 + ... + 3642 = 33662.

(1 + 2 + ... + 10)(11 + 12 + ... + 19)(20 + 21 + ... + 244) = 148502,
(1 + 2 + ... + 23)(24 + 25 + ... + 123)(124 + 125 + ... + 244) = 2125202,
(1 + 2 + ... + 27)(28 + 29 + ... + 196)(197 + 198 + ... + 244) = 2751842,
(1 + 2 + ... + 92)(93 + 94 + ... + 123)(124 + 125 + ... + 244) = 5646962.

(13 + 23 + ... + 153)(163 + 173 + ... + 1193)(1203 + 1213 + ... + 2443) = 248648400002.

2442 = 59536 appears in the decimal expression of e:
  e = 2.71828•••59536••• (from the 14579th digit).

Page of Squares : First Upload October 4, 2004 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

245

The smallest squares containing k 245's :
245025 = 4952,
42451245369 = 2060372,
24524532450625 = 49522252.

1 / 245 = 0.00408163265306122...,
42 + 02 + 82 + 12 + 62 + 32 + 22 + 62 + 52 + 32 + 02 + 62 + 12 + 22 + 22 = 245.

2452 = 60025, 602 + 02 + 252 = 652,
2452 = 60025, 600 + 25 = 252.

2452 = 72 + 82 + 92 + ... + 562.

2452 = (18 + 19 + 20 + 21 + 22 + 23 + 24)2 + (25 + 26 + 27 + 28 + 29 + 30 + 31)2.

32 + 142 + 252 + 362 + 472 + 582 + 692 + 802 + 912 + 1022 + 1132 + 1242 + 1352 + 1462 + 1572 + 1682 + 1792 + 1902 + 2012 + 2122 + 2232 + 2342 + 2452 = 6902.

173k + 245k + 397k + 629k are squares for k = 1,2,3 (382, 8022, 182022).

2452 = 30 + 31 + 35 + 36 + 310.

Komachi equation: 2452 = 92 - 872 + 652 * 42 + 32 + 22 - 102.

3-by-3 magic squares consisting of different squares with constant 2452:

12211522162
14121802882
20021202752
      
20210822192
16521562922
18021552602
      
2427522322
13222002512
20521202602
      
45212822042
16021712722
180212021152

(1 + 2 + ... + 15)(16 + 17 + ... + 240)(241 + 242 + ... + 245) = 648002,
(1 + 2 + ... + 50)(51 + 52 + ... + 60)(61 + 62 + ... + 245) = 1415252,
(1 + 2 + ... + 72)(73 + 74 + ... + 192)(193 + 194 + ... + 245) = 6964202,
(1 + 2 + ... + 90)(91 + 92 + ... + 154)(155 + 156 + ... + 245) = 7644002.

2452 = 60025 appears in the decimal expressions of π and e:
  π = 3.14159•••60025••• (from the 26425th digit),
  e = 2.71828•••60025••• (from the 91044th digit).

Page of Squares : First Upload October 12, 2004 ; Last Revised August 26, 2011
by Yoshio Mimura, Kobe, Japan

246

The smallest squares containing k 246's :
24649 = 1572,
2464824609 = 496472,
20324624624656 = 45082842.

2462 = 60516, a zigzag square.

2462 = 60516, 60 + 5 + 16 = 92,
2462 = 60516, 60 + 516 = 242.

The square sum of the divisors of 246 is 2902.

Komachi equation: 2462 = 12 * 3 + 4 * 5 * 6 * 7 * 8 * 9.

2462 + 2472 + 2482 + 2492 + ... + 15192 = 341252.

(1 + 2 + 3 + 4)(5 + 6 + ... + 59)(60 + 61 + ... + 246) = 224402,
(1 + 2 + ... + 6)(7 + 8 + ... + 14)(15 + 16 + ... + 246) = 73082,
(1 + 2 + ... + 6)(7 + 8 + ... + 110)(111 + 112 + ... + 246) = 556922,
(1 + 2 + ... + 6)(7 + 8 + ... + 201)(202 + 203 + ... + 246) = 655202,
(1 + 2 + ... + 8)(9 + 10 + ... + 110)(111 + 112 + ... + 246) = 728282,
(1 + 2 + ... + 13)(14 + 15 + ... + 77)(78 + 79 + ... + 246) = 851762,
(1 + 2 + ... + 15)(16 + 17 + ... + 201)(202 + 203 + ... + 246) = 1562402,
(1 + 2 + ... + 17)(18 + 19 + ... + 50)(51 + 52 + ... + 246) = 706862,
(1 + 2 + ... + 20)(21 + 22 + ... + 68)(69 + 70 + ... + 246) = 1121402,
(1 + 2 + ... + 48)(49 + 50)(51 + 52 + ... + 246) = 582122,
(1 + 2 + ... + 48)(49 + 50 + ... + 59)(60 + 61 + ... + 246) = 1413722,
(1 + 2 + ... + 63)(64 + 65 + ... + 91)(92 + 93 + ... + 246) = 3385202,
(1 + 2 + ... + 63)(64 + 65 + ... + 125)(126 + 127 + ... + 246) = 5155922,
(1 + 2 + ... + 98)(99 + 100 + ... + 121)(122 + 123 + ... + 246) = 5313002,
(1 + 2 + ... + 105)(106 + 107 + ... + 230)(231 + 232 + ... + 246) = 6678002,
(1 + 2 + ... + 121)(122 + 123 + ... + 241)(242 + 243 + ... + 246) = 4428602,
(1 + 2 + ... + 134)(135 + 136 + ... + 201)(202 + 203 + ... + 246) = 10130402,
(1 + 2 + ... + 160)(161 + 162 + ... + 183)(184 + 185 + ... + 246) = 8307602,
(1 + 2 + ... + 161)(162 + 163 + ... + 229)(230 + 231 + ... + 246) = 8375222.

(12 + 22 + ... + 72)(82 + 92 + ... + 622)(632 + 642 + ... + 2462) = 74736202.

Page of Squares : First Upload October 12, 2004 ; Last Revised May 18, 2010
by Yoshio Mimura, Kobe, Japan

247

The smallest squares containing k 247's :
247009 = 4972,
2472476176 = 497242,
24792472473616 = 49792042.

2472 = 61009, 6 + 1 + 0 + 0 + 9 = 42.

52k + 91k + 117k + 169k + 247k are squares (262, 3382, 47322, 692902) for k = 1, 2, 3, 4.

2472 = 61009 is an exchangeable square. (96100 = 3102).

2472 + 2482 + 2492 + 2502 + ... + 363102 = 39947322.

3-by-3 magic squares consisting of different squares with constant 2472:

329022302
13021952782
21021222452
      
1327822342
16221742672
18621572422
      
18214122022
15821622992
189212221022

(1 + 2 + ... + 66)(67 + 68 + ... + 87)(88 + 89 + ... + 247) = 3095402.

2472 = 61009 appears in the decimal expressions of π and e:
  π = 3.14159•••61099••• (from the 92033rd digit),
  e = 2.71828•••61099••• (from the 125806th digit).

Page of Squares : First Upload October 12, 2004 ; Last Revised January 20, 2009
by Yoshio Mimura, Kobe, Japan

248

The smallest squares containing k 248's :
248004 = 4982,
24824838481 = 1575592,
248124810912484 = 157519782.

1 / 248 = 0.00403225..., 403225 = 6352.

248 = (12 + 22 + 32 + ... + 152) / (12 + 22).

2482 = 61504, a zigzag square with different digits.

2482 = 61504, 6 + 1 + 5 + 0 + 4 = 42,
2482 = 61504, 6 + 15 + 0 + 4 = 52.

1798k + 6882k + 25110k + 27714k are squares for k = 1,2,3 (2482, 380682, 61196482).
2790k + 4278k + 20274k + 34162k are squares for k = 1,2,3 (2482, 400522, 69499522).

Komachi Square Sum : 2482 = 32 + 52 + 62 + 82 + 192 + 2472.

248 is the first integer which is the sum of a square and a prime in 7 ways :
  32 + 239, 52 + 223, 72 + 199, 92 + 167, 112 + 127, 132 + 79, 152 + 23.

2482 + 2492 + 2502 + 2512 + ... + 3432 = 29082.

(1 + 2 + 3)(4 + 5 + ... + 192)(193 + 194 + ... + 248) = 370442,
(1 + 2 + ... + 26)(27 + 28 + ... + 63)(64 + 65 + ... + 248) = 1298702,
(1 + 2 + ... + 27)(28 + 29 + ... + 35)(36 + 37 + ... + 248) = 536762,
(1 + 2 + ... + 27)(28 + 29 + ... + 167)(168 + 169 + ... + 248) = 2948402,
(1 + 2 + ... + 27)(28 + 29 + ... + 192)(193 + 194 + ... + 248) = 2910602,
(1 + 2 + ... + 45)(46 + 47 + ... + 234)(235 + 236 + ... + 248) = 3042902,
(1 + 2 + ... + 50)(51 + 52 + ... + 237)(238 + 239 + ... + 248) = 3029402,
(1 + 2 + ... + 67)(68 + 69 + ... + 220)(221 + 222 + ... + 248) = 5740562,
(1 + 2 + ... + 116)(117 + 118 + ... + 128)(129 + 130 + ... + 248) = 4750202,
(1 + 2 + ... + 134)(135 + 136 + ... + 153)(154 + 155 + ... + 248) = 6874202,
(1 + 2 + ... + 135)(136 + 137 + ... + 203)(204 + 205 + ... + 248) = 10373402,
(1 + 2 + ... + 168)(169 + 170 + ... + 192)(193 + 194 + ... + 248) = 8714162,
(1 + 2 + ... + 215)(216 + 217 + ... + 224)(225 + 226 + ... + 248) = 5108402.

2482 = 61504 appears in the decimal expressions of π and e:
  π = 3.14159•••61504••• (from the 46397th digit),
  e = 2.71828•••61504••• (from the 40259th digit).

Page of Squares : First Upload October 12, 2004 ; Last Revised February 23, 2011
by Yoshio Mimura, Kobe, Japan

249

The smallest squares containing k 249's :
3249 = 572,
249861249 = 158072,
2495249892496 = 15796362.

The squares which begin with 249 and end in 249 are
249861249 = 158072,   2494303249 = 499432,   24903049249 = 1578072,
24945991249 = 1579432,   24982015249 = 1580572,...

2492 = 62001, 6 + 2 + 0 + 0 + 1 = 32,
2492 = 62001, 63 + 23 + 03 + 03 + 13 = 152.

2492 is the second square(> 1) that can not be a sum of a power of 2 and a prime.

4233k + 13944k + 16434k + 27390k are squares for k = 1,2,3 (2492, 351092, 52700852).

3-by-3 magic squares consisting of different squares with constant 2492:

824922442
7922322442
2362762232
    
1624722442
6822362412
2392642282
    
1628922322
16721762562
18421522712
    
20212422152
17521602762
176214521002
    
41212422122
14821842792
196211321042

(1 + 2)(3 + 4 + 5 + 6)(7 + 8 + ... + 249) = 12962,
(1 + 2 + ... + 35)(36 + 37 + ... + 39)(40 + 41 + ... + 249) = 535502,
(1 + 2 + ... + 62)(63 + 64 + ... + 93)(94 + 95 + ... + 249) = 3554462,
(1 + 2 + ... + 62)(63 + 64 + ... + 153)(154 + 155 + ... + 249) = 6093362,
(1 + 2 + ... + 71)(72 + 73 + ... + 142)(143 + 144 + ... + 249) = 6381482.

2492 = 62001 appears in the decimal expression of e:
  e = 2.71828•••62001••• (from the 91511st digit).

Page of Squares : First Upload October 12, 2004 ; Last Revised February 23, 2011
by Yoshio Mimura, Kobe, Japan