330
The smallest squares containing k 330's :
330625 = 5752,
330330625 = 181752,
330533307330625 = 181805752.
330k + 1353k + 4026k + 4092k are squares for k = 1,2,3 (992, 59072, 3691712).
13090k + 23210k + 31130k + 41470k are squares for k = 1,2,3 (3302, 583002, 107811002).
10k + 50k + 190k + 320k + 330k are squares (302, 5002, 87002, 1538002) for k = 1, 2, 3, 4.
The integral triangle of sides 520, 641, 1089 has square area 3302.
3302 = 153 + 373 + 383.
3302 = (32 + 6)(42 + 6)(182 + 6).
3302 = 108900 appears in the decimal expression of π:
π = 3.14159•••108900••• (from the 102869th digit).
by Yoshio Mimura, Kobe, Japan
331
The smallest squares containing k 331's :
33124 = 1822,
23313319969 = 1526872,
143313316593316 = 119713542.
3312 = 109561, a zigzag square.
Komachi equation: 3312 = 12 * 22 * 342 * 52 - 62 - 782 + 92.
3-by-3 magic squares consisting of different squares with constant 3312:
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3312 = 109561, 12 + 02 + 92 + 52 + 62 + 12 = 122,
3312 = 109561, 109 + 5 + 6 + 1 = 112,
3312 = 109561, 1095 + 61 = 342.
3312 + 3322 + 3332 + 3342 + 3352 + ... + 180432 = 13993272.
Page of Squares : First Upload December 13, 2004 ; Last Revised May 28, 2010by Yoshio Mimura, Kobe, Japan
332
The smallest squares containing k 332's :
133225 = 3652,
3323329 = 18232,
333267332983321 = 182556112.
332 is the second integer which is the sum of a square and a prime in 8 ways :
12 + 331, 52 + 307, 72 + 283, 92 + 251, 112 + 211, 132 + 163, 152 + 107, 172 + 43.
Komachi equation: 3322 = 92 * 82 - 72 + 62 * 542 + 32 + 22 + 102.
3322 = (12 + 22 + ... + 282) + (12 + 22 + ... + 672).
3323329 = 18232.
34652 = 912 + 922 + 932 + 942 + 952 + ... + 3322.
Page of Squares : First Upload December 13, 2004 ; Last Revised May 28, 2010by Yoshio Mimura, Kobe, Japan
333
The smallest squares containing k 333's :
1833316 = 13542,
3333330225 = 577352,
3833363332333921 = 619141612.
1554k + 23865k + 37740k + 47730k are squares for k = 1,2,3 (3332, 653792, 132697172). 3108k + 10212k + 39072k + 58497k are squares for k = 1,2,3 (3332, 711512,161528312). 6216k + 18426k + 34521k + 51726k are squares for k = 1,2,3 (3332, 651572, 136393472). 17094k + 19092k + 32190k + 42513k are squares for k = 1,2,3 (3332, 591632, 110519372).
3-by-3 magic squares consisting of different squares with constant 3332:
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3332 = 110889, 11 + 0 + 8 + 8 + 9 = 62,
3332 = 110889, 11 + 0 + 889 = 302.
3332 = 110889 appears in the decimal expression of π:
π = 3.14159•••110889••• (from the 75391st digit).
by Yoshio Mimura, Kobe, Japan
334
The smallest squares containing k 334's :
33489 = 1832,
43342243344 = 2081882,
39334233433401 = 62717012.
3342 = 111556, a square with 3 kinds of digits and a non - decreasing sequence of digits.
3342 = 111556, a square with odd digits except the last digit 6.
Komachi equation: 3342 = 12 * 232 / 42 * 562 - 72 + 892.
3342 = 111556, 1 + 1 + 1 + 5 + 56 = 82,
3342 = 111556, 1 + 1 + 1 + 55 + 6 = 82.
3342 = 111556 appears in the decimal expression of e:
e = 2.71828•••111556••• (from the 123336th digit)
by Yoshio Mimura, Kobe, Japan
335
The smallest squares containing k 335's :
335241 = 5792,
33353351641 = 1826292,
3350133533569 = 18303372.
The square root of 335 is 18.303..., 18 = 32 + 02 + 32.
3352 = 112225, a square with 3 kinds of digits and a non - decreasing sequence of digits.
Komachi Square Sums : 3352 = 42 + 82 + 762 + 952 + 3122.
(12 + 22 + ... + 3352) = 12587960, which consists of different digits,
This is the first 8-digit sum (there are 6 sums in all).
3-by-3 magic squares consisting of different squares with constant 3352:
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3352 = 112225, 1 + 1 + 22 + 25 = 72,
3352 = 112225, 112 + 2 + 2 + 5 = 112.
(13 + 23 + ... + 203)(213 + 223 + ... + 1043)(1053 + 1063 + ... + 3353) = 641787300002,
(13 + 23 + ... + 393)(403 + 413 + ... + 1043)(1053 + 1063 + ... + 3353) = 2361078720002,
(13 + 23 + ... + 603)(613 + 623 + ... + 1043)(1053 + 1063 + ... + 3353) = 5273134020002.
by Yoshio Mimura, Kobe, Japan
336
The smallest squares containing k 336's :
3364 = 582,
1336336 = 11562,
63362523363364 = 79600582.
The squares which begin with 336 and end in 336 are
336502336 = 183442, 33615022336 = 1833442, 336001078336 = 5796562,
336219064336 = 5798442, 336580984336 = 5801562,...
The square root of 336 is 18.33..., 18 = 32 + 32.
3362 = 1122 + 2242 + 2242 : 4222 + 4222 + 2112 = 6332.
3362 = (12 + 3)(32 + 3)(52 + 3)(92 + 3) = (22 - 1)(132 - 1)(152 - 1) = (32 - 1)(82 - 1)(152 - 1).
The integral triangle of sides 1009, 3088, 4095 has square area 3362.
Cubic Polynomial : (X + 72)(X + 1082)(X + 3362) = X3 + 3532X2 + 363722X + 2540162.
(X + 3362)(X + 16842)(X + 107732) = X3 + 109092X2 + 185079722X + 60956219522.
54 + 336 = 312, 54 - 336 = 172.
Komachi Fraction : 675 / 3048192 = (5 / 336)2.
Komachi equations:
3362 = 1 + 2 - 3 + 4 * 56 * 7 * 8 * 9 = - 1 - 2 + 3 + 4 * 56 * 7 * 8 * 9,
3362 = 12 / 22 / 32 * 42 * 5672 * 82 / 92 = 12 * 22 * 342 * 52 - 62 * 782 / 92
= 92 + 82 * 72 * 62 - 542 / 32 / 22 * 12 = 92 + 82 * 72 * 62 - 542 / 32 / 22 / 12
= 92 * 82 * 72 * 62 / 542 * 32 * 22 * 12 = 92 * 82 * 72 * 62 / 542 * 32 * 22 / 12
= 92 * 82 * 72 / 62 * 52 - 42 * 32 * 212 = 982 / 72 * 62 * 52 - 42 * 32 * 212
= - 92 + 82 * 72 * 62 + 542 / 32 / 22 * 12 = - 92 + 82 * 72 * 62 + 542 / 32 / 22 / 12
= - 92 + 872 + 62 * 542 - 32 + 212 = 92 * 82 * 72 / 62 * 52 / 42 * 322 / 102
= 92 / 82 * 72 / 62 / 52 * 42 * 322 * 102 = 982 / 72 * 62 * 52 / 42 * 322 / 102
= - 982 + 72 * 62 * 52 / 42 / 32 * 22 * 102 = - 982 + 72 / 62 * 52 * 42 * 32 / 22 * 102
= - 982 + 72 * 62 * 52 + 42 / 32 * 2102 = - 982 + 72 * 62 * 52 * 42 / 32 + 2102.
1336336 = 11562.
3362 = 112896, 1 + 1 + 2 + 896 = 302,
3362 = 112896, 1 + 12 + 8 + 9 + 6 = 62,
3362 = 112896, 1 + 128 + 9 + 6 = 122,
3362 = 112896, 1 + 128 + 96 = 152,
3362 = 112896, 1 + 1289 + 6 = 362,
3362 = 112896, 11 + 2 + 8 + 9 + 6 = 62.
22052 = 2872 + 2882 + 2892 + 2902 + 2912 + ... + 3362.
(12 + 22 + 32)(42 + 52 + 62 + 72)(82) = 3362.
3362 = 112896 appears in the decimal expressions of π and e:
π = 3.14159•••112896••• (from the 96315th digit),
e = 2.71828•••112896••• (from the 131340th digit).
by Yoshio Mimura, Kobe, Japan
337
The smallest squares containing k 337's :
337561 = 5812,
13543373376 = 1163762,
1337995733733376 = 365786242.
The square root of 337 is 18.3575597506...,
182 = 32 + 52 + 72 + 52 + 52 + 92 + 72 + 52 + 02 + 62.
3372 = 113569, a square with a non - decreasing sequence of digits.
3372 = 232 + 1922 + 2762 : 6722 + 2912 + 322 = 7332.
3372 = 94 + 124 + 124 + 164.
13 + 33 + 53 + 73 + 93 + ... + 3373 = 403912.
Cubic Polynomial : (X + 1052)(X + 1402)(X + 2882) = X3 + 3372X2 + 525002X + 42336002.
Komachi equation: 3372 = 1 + 234 * 56 * 78 / 9.
3372 = 113569, 1 + 1 + 3 + 5 + 6 + 9 = 52.
3-by-3 magic squares consisting of different squares with constant 3372:
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by Yoshio Mimura, Kobe, Japan
338
The smallest squares containing k 338's :
33856 = 1842,
33806338225 = 1838652,
473380233833809 = 217573032.
3382 = 114244, a square with 3 kinds of digits.
3382 = (12 + 1)(2392 + 1).
26k + 338k + 910k + 1226k are squares for k = 1,2,3 (502, 15642, 513322).
190k + 338k + 786k + 802k are squares for k = 1,2,3 (462, 11882, 323562).
Komachi Fraction : 3382 = 9253764 / 81.
A + B, A + C, A + D, B + C, B + D, and C + D are squares for A = 338, B = 1106, C = 3383, D = 8498.
3382 = 134 + 134 + 134 + 134.
3382 = 114244, 1 + 1 + 4 + 2 + 4 + 4 = 42,
3382 = 114244, 1 + 14 + 2 + 4 + 4 = 52,
3382 = 114244, 11 + 4 + 2 + 4 + 4 = 52,
3382 = 114244, 112 + 42 + 242 + 42 = 272.
by Yoshio Mimura, Kobe, Japan
339
The smallest squares containing k 339's :
133956 = 3662,
23393396601 = 1529492,
339453393396121 = 184242612.
The square root of 339 is 18.411..., 18 = 42 + 12 + 12.
3842k + 26216k + 34465k + 50398k are squares for k = 1,2,3 (3392, 665572, 136755992).
3392 = 1132 + 2262 + 2262 : 6222 + 6222 + 3112 = 9332.
Komachi Fraction : 576 / 1034289 = (8 / 339)2.
3-by-3 magic squares consisting of different squares with constant 3392:
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3394 = 13206836241, 132 + 22 + 02 + 62 + 82 + 32 + 62 + 22 + 42 + 12 = 339.
3392 = 114921, 1 + 1 + 4 + 9 + 21 = 62,
3392 = 114921, 11 + 49 + 21 = 92,
3392 = 114921, 114 + 9 + 21 = 122.
(13 + 23 + ... + 483)(493 + 503 + ... + 3393) = 677587682,
(13 + 23 + ... + 3323)(3333 + 3343 + ... + 3393) = 9008102882.
by Yoshio Mimura, Kobe, Japan