The square of 6.
The smallest squares containing k 36's :
36 = 62, 93636 = 3062, 1036453636 = 321942,
3636813636 = 603062, 36360369361936 = 60299562.
The squares which begin with 36 and end in 36 are
367236 = 6062, 3632836 = 19062, 36072036 = 60062, 36529936 = 60442,
36675136 = 60562,...
A Kaprekar number : 362 = 1296, 1 + 29 + 6 = 36.
36 is the sum of m squares for m = 3, 4, ..., 22.
362± 5 are primes.
The 3rd integer which is the sum of 4 squares in just 2 ways: [1,1,3,5], [3,3,3,3].
The 1st integer which is the sum of 6 squares in just 4 ways:
[1,1,1,1,4,4], [1,1,1,2,2,5], [1,2,2,2,3,3], [2,2,2,2,2,4].
362 = 23 + 63 + 73 + 93.
362 = 122 + 242 + 242, 422 + 422 + 212 = 632.
362 = 1296, its digits being distinct.
362 = (22 + 8)(102 + 8).
362 + 372 + 382 + 392 + 402 = 412 + 422 + 432 + 442.
93636 = 3062, 367236 = 6062.
253k + 295k + 341k + 407k are squares for k = 1,2,3 (362, 6582, 122042).
6k + 12k + 36k + 129k + 258k are squares 212, 2912, 44012, 686252 for k = 1, 2, 3, 4.
Cubic polynomial (X + 362)(X + 4272)(X + 6722) = X3 + 7972X2 + 2883722X + 103299842.
Komachi Fractions : 362 = 361584/279, (5/36)2 = 6075/314928.
Komachi equations:
362 = 12 * 3 * 4 * 56 / 7 / 8 * 9 = 12 * 3 * 4 / 56 * 7 * 8 * 9
= 1234 + 56 + 7 + 8 - 9, and more 3 equations,
362 = 9 - 8 + 7 + 6 * 5 * 43 - 2 * 1 = 9 * 8 * 7 * 6 - 54 * 32 * 1
= 98 + 765 + 432 + 1, and more 4 equations,
362 = 987 + 6 * 54 - 3 - 2 - 10 = 987 + 6 * 54 - 3 / 2 * 10
= 9 * 8 * 7 + 65 * 4 * 3 + 2 + 10, and more 9 equations,
362 = 12 * 22 * 32 * 42 + 52 - 672 + 82 * 92 = 12 + 22 - 32 / 42 * 562 + 72 * 82 - 92,
362 = 92 + 82 + 72 + 62 + 52 + 42 + 322 + 12 = 982 / 72 + 62 + 52 + 42 + 322 - 12,
362 = 123 - 33 + 43 - 53 + 63 - 73 + 83 - 93,
362 = 14 / 24 / 34 * 44 * 564 / 74 / 84 * 94 = 14 / 24 / 34 * 44 / 564 * 74 * 84 * 94,
362 = 94 + 84 + 74 - 64 - 54 + 44 - 34 - 24 - 104 = 94 * 84 * 74 * 64 * 54 / 44 / 34 / 2104
= 94 * 84 / 74 / 64 / 54 / 44 / 34 * 2104.
(362 + 4) = (12 + 4)(162 + 4) = (12 + 4)(32 + 4)(42 + 4),
(362 + 6) = (52 + 6)(62 + 6), (362 - 8) = (42 - 8)(132 - 8),
(362 - 6) = (62 - 6)(72 - 6) = (32 - 6)(42 - 6)(72 - 6).
362 = (1 + 2)(3 + 4 + 5 + 6)(7 + 8 + 9) = (1 + 2)(3 + 4 + ... + 29) = (1)(2 + 3 + 4)(5 + 6 + 7)(8),
362 = 13 + 23 + 33 + 43 + 53 + 63 + 73 + 83.
(1 + 2 + 3)(4 + 5 + ... + 12)(13 + 14 + ... + 36) = 5042,
(1 + 2 + ... + 5 )( 6 + 7 + ... + 26)(27 + 28 + ... + 36) = 12602,
(1 + 2 + ... + 7 )( 8 + 9 + ... + 27)(28 + 29 + ... + 36) = 16802,
(1 + 2 + ... + 8 )( 9 + 10 + ... + 26)(27 + 28 + ... + 36) = 18902,
(1 + 2 + ... + 13)(14 + 15 + ... + 26)(27 + 28 + ... + 36) = 27302,
(1 + 2 + ... + 13)(14 + 15 + ... + 28)(29 + 30 + ... + 36) = 27302.
(13 + 23 + ... + 273)(283 + 293 + ... + 353)(363) = 411505922.
13 + 23 + ... + 363 = (1 + 2 + ... 36)2 = 6662.
362 = 1296, 1 + 29 + 6 = 62,
364 = 1679616, 1 + 679 + 616 = 362, 366 = 2176782336, 2 + 176 + 782 + 336 = 362,
368 = 2821109907456, 2 + 8 + 2 + 1109 + 90 + 74 + 5 + 6 = 362,
2 + 8 + 211 + 0990 + 74 + 5 + 6 = 28 + 2 + 1109 + 90 + 7 + 4 + 56
= 28 + 211 + 0990 + 7 + 4 + 56 = 282 + 1 + 1 + 0990 + 7 + 4 + 5 + 6 = 362,
369 = 101559956668416, 1 + 01 + 5 + 5 + 9 + 9 + 566 + 684 + 16 = 362,
1 + 01 + 5 + 599 + 5 + 6 + 668 + 4 + 1 + 6 = 1 + 01 + 5 + 599 + 5 + 666 + 8 + 4 + 1 + 6
= 1 + 01 + 55 + 99 + 56 + 668 + 416 = 1 + 01 + 559 + 9 + 5 + 6 + 668 + 41 + 6
= 1 + 01 + 559 + 9 + 5 + 666 + 8 + 41 + 6 = 1 + 015 + 5 + 9 + 9 + 566 + 684 + 1 + 6
= 1 + 015 + 599 + 566 + 68 + 41 + 6 = 1 + 0155 + 995 + 66 + 68 + 4 + 1 + 6
= 10 + 1 + 5 + 5 + 9 + 9 + 566 + 684 + 1 + 6 = 10 + 1 + 5 + 599 + 566 + 68 + 41 + 6
= 10 + 1 + 559 + 9 + 5 + 6 + 6 + 684 + 16 = 10 + 15 + 599 + 566 + 6 + 84 + 16
= 10 + 155 + 9 + 956 + 66 + 84 + 16 = 101 + 5 + 599 + 566 + 6 + 8 + 4 + 1 + 6
= 101 + 55 + 995 + 66 + 68 + 4 + 1 + 6 = 101 + 559 + 9 + 566 + 6 + 8 + 41 + 6
= 1015 + 5 + 9 + 95 + 6 + 66 + 84 + 16 = 1015 + 5 + 9 + 95 + 66 + 6 + 84 + 16
= 1015 + 5 + 99 + 5 + 6 + 66 + 84 + 16 = 1015 + 5 + 99 + 5 + 66 + 6 + 84 + 16
= 1015 + 5 + 99 + 56 + 6 + 68 + 41 + 6 = 1015 + 5 + 99 + 56 + 66 + 8 + 41 + 6
= 1015 + 59 + 9 + 56 + 66 + 84 + 1 + 6 = 1015 + 59 + 95 + 6 + 6 + 68 + 41 + 6
= 1015 + 59 + 95 + 6 + 66 + 8 + 41 + 6 = 1015 + 59 + 95 + 66 + 6 + 8 + 41 + 6 = 362.
362 = 1296 appears in the decimal expressions of π and e:
π = 3.14159•••1296••• (from the 2341st digit),
e = 2.71828•••1296••• (from the 4822nd digit).
Page of Squares : First Upload December 1, 2003 ; Last Revised January 13, 2014
by Yoshio Mimura, Kobe, Japan