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.)
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, 2007by 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.
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, 2014by Yoshio Mimura, Kobe, Japan
4107
41072 = 16867449, 1 * 68 * 67 - 449 = 4107.
Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007by 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).
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, 2010by 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, 2008by 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, 2007by 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, 2007by 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, 2013by 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, 2007by 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, 2007by 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, 2011by Yoshio Mimura, Kobe, Japan
4125
41252 = 123 + 343 + 2573.
Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008by Yoshio Mimura, Kobe, Japan
4126
41262± 3 are primes.
Page of Squares : First Upload January 18, 2014 ; Last Revised January 18, 2014by Yoshio Mimura, Kobe, Japan
4128
41282 = 2403 + 205 + 47.
Page of Squares : First Upload October 8, 2010 ; Last Revised October 8, 2010by 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, 2013by 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, 2008by 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, 2007by Yoshio Mimura, Kobe, Japan
4137
41372 = 723 + 1403 + 2413.
Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008by 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, 2008by 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, 2007by 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, 2007by Yoshio Mimura, Kobe, Japan
4144
41442 = 17172736, a zigzag square.
Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007by 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, 2008by Yoshio Mimura, Kobe, Japan
4149
41492 = 165 + 205 + 205 + 255.
Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008by 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, 2014by 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, 2008by Yoshio Mimura, Kobe, Japan
4152
41522 = 1063 + 1673 + 2253.
Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008by 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, 2008by Yoshio Mimura, Kobe, Japan
4155
41552 = 563 + 1773 + 2263.
Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008by 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, 2007by 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, 2011by 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, 2008by 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, 2010by 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)
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, 2008by 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, 2011by 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, 2013by 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, 2014by Yoshio Mimura, Kobe, Japan
4169
41692 = (32 + 2)(12572 + 2).
Page of Squares : First Upload December 28, 2013 ; Last Revised December 28, 2013by 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, 2007by 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, 2007by 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, 2007by 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, 2007by Yoshio Mimura, Kobe, Japan
4175
41752 = 17430625, a square with different digits.
Page of Squares : First Upload August 20, 2007 ; Last Revised August 20, 2007by 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, 2008by 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, 2013by 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, 2008by 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, 2008by Yoshio Mimura, Kobe, Japan
4182
41822 = (22 + 2)(242 + 2)(712 + 2).
Page of Squares : First Upload December 28, 2013 ; Last Revised December 28, 2013by 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, 2008by 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, 2007by 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, 2011by Yoshio Mimura, Kobe, Japan
4191
41912 = 873 + 1033 + 2513.
Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008by Yoshio Mimura, Kobe, Japan
4192
41922 = 25 + 145 + 225 + 265.
Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008by 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, 2007by 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, 2007by 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, 2007by 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, 2011by Yoshio Mimura, Kobe, Japan