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4100 - 4199

4100

41002 = 16810000, and 16 = 42, 810000 = 9002.

4-cycle : 41002 = 16810000 -- 81002 = 65610000 -- 61002 = 37210000 -- 21002 = 04410000,
(Other examples : 2916 -- 5030 -- 3009 -- 0540 -- 2916, 1600 -- 5600 -- 3600 -- 9600 -- 1600.)

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4102

41022 = 16826404, 1 * 682 * 6 + 40 / 4 = 4102.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4104

41042 = (72 + 8)(122 + 8)(442 + 8) = (72 + 8)(82 + 8)(642 + 8).

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

Komachi equations:
41042 = 92 * 82 * 762 * 52 / 42 * 32 * 22 / 102 = 92 * 82 * 762 / 52 / 42 * 32 / 22 * 102
 = 92 / 82 * 762 / 52 * 42 * 32 * 22 * 102.

Page of Squares : First Upload August 20, 2007 ; Last Revised December 28, 2013
by Yoshio Mimura, Kobe, Japan

4106

41062± 3 are primes.

41062 = 16859236, 12 + 62 + 82 + 52 + 92 + 22 + 32 + 62 = 162.

Page of Squares : First Upload August 20, 2007 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

4107

41072 = 16867449, 1 * 68 * 67 - 449 = 4107.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4108

2080k + 3497k + 4004k + 4108k are squares for k = 1,2,3 (1172, 70332, 4304432).

4108 = (12 + 22 + 32 + ... + 392) / (12 + 22),
4108 = (12 + 22 + 32 + ... + 5522) / (12 + 22 + 32 + ... + 342).

Page of Squares : First Upload November 25, 2008 ; Last Revised June 10, 2011
by Yoshio Mimura, Kobe, Japan

4109

1 / 4109 = 0.00024336...., and 24336 = 1562.

41092 = S2(286) + S2(300), where S2(n) = 12 + 22 + 32 + 42 + 52 + ... + n2.

Komachi equation: 41092 = 982 * 72 * 62 + 52 - 42 * 32 * 22 * 102.

Page of Squares : First Upload August 20, 2007 ; Last Revised October 8, 2010
by Yoshio Mimura, Kobe, Japan

4112

41122 = 633 + 1213 + 2463 = 44 + 164 + 164 + 644 = 28 + 48 + 48 + 88.

Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

4113

41132 = 16916769, 1 * 691 * 6 - 7 * 6 + 9 = 4113.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4115

41152 = 16933225, 12 + 62 + 92 + 32 + 32 + 22 + 22 + 52 = 132.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4116

41162 = 443 + 1503 + 2383.

41162 = (22 + 3)(32 + 3)(122 + 3)(372 + 3) = (92 + 3)(122 + 3)(372 + 3).

Komachi equations: 41162 = 982 * 72 / 62 * 542 / 32 * 22 */ 12.

41162 = (11 + 12 + 13 + ... + 38)2 + (39 + 40 + 41 + ... + 66)2 + (67 + 68 + 69 + ... + 94)2 + ... + (95 + 96 + 97 + ... + 122)2.

Page of Squares : First Upload August 20, 2007 ; Last Revised December 28, 2013
by Yoshio Mimura, Kobe, Japan

4118

41182 = S2(229) + S2(338), where S2(n) = 12 + 22 + 32 + 42 + 52 + ... + n2.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4119

41192 = 16966161, a square with just 3 kinds of digits.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4122

786k + 1614k + 3882k + 4122k are squares for k = 1,2,3 (1022, 59402, 3650042).

Page of Squares : First Upload June 10, 2011 ; Last Revised June 10, 2011
by Yoshio Mimura, Kobe, Japan

4125

41252 = 123 + 343 + 2573.

Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

4126

41262± 3 are primes.

Page of Squares : First Upload January 18, 2014 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

4128

41282 = 2403 + 205 + 47.

Page of Squares : First Upload October 8, 2010 ; Last Revised October 8, 2010
by Yoshio Mimura, Kobe, Japan

4130

41302 = (22 + 6)(172 + 6)(762 + 6).

41302 = 15 + 95 + 155 + 185 + 275.

Page of Squares : First Upload August 26, 2008 ; Last Revised December 28, 2013
by Yoshio Mimura, Kobe, Japan

4131

41312 = 93 + 1623 + 2343 = 653 + 1963 + 2103.

41312 = 17065161, 170 * 6 + 51 * 61 = 4131.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

4132

41322 = 17073424, a zigzag square.

41322 = 17073424, 12 + 72 + 02 + 72 + 32 + 42 + 22 + 42 = 122.

41322 = 17073424, 1 + 707 + 3424 = 4132.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4137

41372 = 723 + 1403 + 2413.

Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

4138

41385 = 1213255955047723168 : 12 + 22 + 12 + 32 + 22 + 52 + 52 + 92 + 552 + 02 + 42 + 72 + 72 + 232 + 162 + 82 = 12 + 22 + 12 + 32 + 22 + 552 + 92 + 52 + 52 + 02 + 42 + 72 + 72 + 232 + 162 + 82 = 122 + 12 + 32 + 252 + 52 + 92 + 552 + 02 + 42 + 72 + 72 + 22 + 32 + 12 + 62 + 82 = 4138.

Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

4140

S2(4140) = S2(3170) + S2(3394), where S2(n) = 12 + 22 + 32 + 42 + 52 + ... + n2.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4141

41412 = 17147881, 1 * 7 - 1 + 47 * 88 - 1 = 4141.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4144

41442 = 17172736, a zigzag square.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4148

4148 = (12 + 22 + 32 + ... + 1522) / (12 + 22 + 32 + ... + 92).

Page of Squares : First Upload November 25, 2008 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

4149

41492 = 165 + 205 + 205 + 255.

Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

4150

41502± 3 are primes.

41502 = 303 + 853 + 2553.

Page of Squares : First Upload August 26, 2008 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

4151

41512 = 17230801, a zigzag square.

41515 = 1232433988374745751 : 12 + 232 + 22 + 432 + 32 + 92 + 82 + 82 + 372 + 42 + 72 + 42 + 52 + 72 + 52 + 12 = 4151.

Page of Squares : First Upload August 20, 2007 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

4152

41522 = 1063 + 1673 + 2253.

Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

4153

41532 = 17247409, 1 * 7 * 2 * 4 * 74 + 0 + 9 = 4153.

41534 = 297473117213281, and 22 + 92 + 72 + 472 + 312 + 12 + 72 + 22 + 12 + 32 + 282 + 12 = 4153.

Page of Squares : First Upload August 20, 2007 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

4155

41552 = 563 + 1773 + 2263.

Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

4157

41572 = 17280649, a zigzag square with different digits.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4158

41582 = 17288964, 1 * 7 / 2 * 88 * 9 * 6 / 4 = 4158.

The integral triangle of sides 11907, 17545, 29354 has square area 41582.

Page of Squares : First Upload August 20, 2007 ; Last Revised October 11, 2011
by Yoshio Mimura, Kobe, Japan

4159

41592 = 603 + 1183 + 2493.

(41592 - 1) = (22 - 1)(52 - 1)(62 - 1)(72 - 1)(122 - 1) = (22 - 1)(32 - 1)(82 - 1)(92 - 1)(122 - 1).

Page of Squares : First Upload August 20, 2007 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

4160

41602 = 46 + 86 + 86 + 166.

Komachi equation: 41602 = 122 / 32 * 42 * 52 * 62 * 782 / 92.

Page of Squares : First Upload August 26, 2008 ; Last Revised October 8, 2010
by Yoshio Mimura, Kobe, Japan

4163

Loop of length 56 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
4163 - 5650 - 5636 - 4432 - ... - 8450 - 9556 - 12161 - 4163
(Note f(4163) = 412 + 632 = 5650,   f(5650) = 562 + 502 = 5636, etc. See 41)

Page of Squares : First Upload October 9, 2008 ; Last Revised October 9, 2008
by Yoshio Mimura, Kobe, Japan

4164

41642 = 183 + 1873 + 2213.

1 / 4164 = 0.0002401...., and 2401 = 492.

41642 = 17338896, 173 * 3 * 8 + 8 * 9 / 6 = 4164.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

4165

4165k + 7973k + 17017k + 27489k are squares for k = 1,2,3 (2382, 335582, 51262822).

Page of Squares : First Upload June 10, 2011 ; Last Revised June 10, 2011
by Yoshio Mimura, Kobe, Japan

4166

41662 = 17355556, a square with odd digits except the last digit 6.

41662 = 1453 + 1703 + 2113.

1 / 4166 = 0.000240038406144983197....,
  and 22 + 4002 + 382 + 40612 + 42 + 492 + 8312 + 972 = 41662.

41662 = 17355556, 1 + 73 * 55 + 5 * 5 * 6 = 4166.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 24, 2013
by Yoshio Mimura, Kobe, Japan

4168

41682± 3 are primes.

41682 = 513 + 1333 + 2463.

1 / 4168 = 0.0002399232245...., 22 + 392 + 92 + 232 + 22 + 22 + 452 = 4168.

Page of Squares : First Upload August 20, 2007 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

4169

41692 = (32 + 2)(12572 + 2).

Page of Squares : First Upload December 28, 2013 ; Last Revised December 28, 2013
by Yoshio Mimura, Kobe, Japan

4170

41702= 366 x 367 + 367 x 368 + 368 x 369 + 369 x 370 + ... + 465 x 466.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4172

41722 = 17405584, 12 + 72 + 42 + 02 + 52 + 52 + 82 + 42 = 142.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4173

41732 = 17413929, 17 * 41 / 3 * 9 * 2 - 9 = 4173.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4174

41742 = 17422276, 1 - 7 + 4222 - 7 * 6 = 4174.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4175

41752 = 17430625, a square with different digits.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4177

Loop of length 56 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
4177 - 7610 - 5876 - ... - 3878 - 7528 - 6409 - 4177
(Note f(4177) = 412 + 772 = 7610,   f(7610) = 762 + 102 = 5876, etc. See 41)

41772 = 94 + 244 + 244 + 644.

41772 = 17447329, 12 + 72 + 42 + 42 + 72 + 32 + 22 + 92 = 152.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

4179

41792 = 17464041, 174 * 6 * 4 + 0 + 4 - 1 = 4179.

41792± 2 are primes.

41792 = 17464041, and 12 + 72 + 42 + 642 + 02 + 42 + 12.

Page of Squares : First Upload August 20, 2007 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan

4180

4180 = (12 + 22 + 32 + ... + 1042) / (12 + 22 + 32 + ... + 62).

Page of Squares : First Upload November 25, 2008 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

4181

41812 = 413 + 623 + 2583.

41814 = 305577005139121, and 302 + 52 + 52 + 72 + 72 + 02 + 02 + 512 + 32 + 92 + 12 + 212 = 4181.

Page of Squares : First Upload August 26, 2008 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

4182

41822 = (22 + 2)(242 + 2)(712 + 2).

Page of Squares : First Upload December 28, 2013 ; Last Revised December 28, 2013
by Yoshio Mimura, Kobe, Japan

4184

41842 = 1603 + 1643 + 2083.

41842 = 17505856, 12 + 72 + 52 + 02 + 52 + 82 + 52 + 62 = 152.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

4186

41862 = 17522596, 12 + 72 + 52 + 22 + 22 + 52 + 92 + 62 = 152.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4190

41902 = 17556100, 1 * 75 * 56 - 10 + 0 = 4190.

730k + 3970 k +4190 k +8010k are squares for k = 1,2,3 (1302, 99002, 8065002).

Page of Squares : First Upload August 20, 2007 ; Last Revised June 10, 2011
by Yoshio Mimura, Kobe, Japan

4191

41912 = 873 + 1033 + 2513.

Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

4192

41922 = 25 + 145 + 225 + 265.

Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

4193

41932 = 17581249, 17 + 58 * 1 * 2 * 4 * 9 = 4193.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4194

41942 = 17589636, 1 * 75 * 8 / 9 * 63 - 6 = 4194.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4196

41962 = 17606416, 1 + 7 * 606 - 41 - 6 = 4196.

Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007
by Yoshio Mimura, Kobe, Japan

4199

408k + 1156k + 2601k + 3060k are squares for k = 1,2,3 (852, 41992, 2187732).

41992 = 93 + 383 + 2603 = 1373 + 1953 + 1973.

Page of Squares : First Upload August 26, 2008 ; Last Revised June 10, 2011
by Yoshio Mimura, Kobe, Japan