The square of 4.
The smallest squares containing k 16's :
16 = 42, 41616 = 2042, 161391616 = 127042,
1616361616 = 402042, 116981616166416 = 108158042.
The squares which begin with 16 and end in 16 are
163216 = 4042, 1679616 = 12962, 16032016 = 40042, 16370116 = 40462,
16434916 = 40542,...
162 = 256 with an increasing sequence of digits: 2, 5, 6
An exchangeable square: 162 = 256, 625 = 252.
2042 = 41616.
162 = 256, 2 * 5 + 6 = 16.
162 = (12 + 7)(52 + 7).
162 = 30 + 31 + 32 + 35.
162 + 172 + 182 + ... + 1422 = 1432 + 1442 + 1452 + ... + 1792.
Loop of length 8 by the function f(N) = ... + c2 + b2 + a2 for N = ... + 102c + 10b + a:
16 -- 37 -- 58 -- 89 -- 145 -- 42 -- 20 -- 4 -- 16
The smallest square that is the sum of two triangular numbers in two ways:
(1 + 2 + 3) + (1 + 2 + 3 + 4) = 1 + (1 + 2 + 3 + 4 + 5)
162 is the first square which is the sum of 5 distinct squares in just 4 ways:
12 + 22 + 72 + 92 + 112 = 12 + 32 + 52 + 102 + 112 = 12 + 52 + 72 + 92 + 102
= 22 + 32 + 52 + 72 + 132.
162 is the first square which is the sum of 6 distinct squares in just 6 ways.
162 is the first square which is the sum of 7 distinct squares in just 2 ways.
162 is the first square which is the sum of 6 cubes in just 2 ways:
13 + 13 + 13 + 43 + 43 + 53 = 23 + 23 + 23 + 23 + 23 + 63.
162 is the 6th square which is the sum of 4 cubes : 43 + 43 + 43 + 43.
162 is the second square which is the sum of 4 sixth powers: 26 + 26 + 26 + 26.
162 is the first square which is the sum of 2 seventh powers: 27 + 27.
Komachi Fractions: (16/289)2 = (4/17)4 = 2304/751689, (3/16)2 = 3726/105984.
Komachi equations:
162 = 1 - 23 - 4 + 5 * 6 * 7 + 8 * 9 = 1 + 2 * 34 * 5 - 6 - 7 - 8 * 9
= 1 + 234 + 5 + 6 - 7 + 8 + 9, and more 42 equations,
162 = 9 - 8 - 7 + 65 * 4 + 3 - 2 + 1 = 98 - 7 + 6 + 54 * 3 - 2 - 1
= 9 + 8 + 76 + 54 * 3 + 2 - 1, and more 28 equations,
162 = 9 * 8 + 7 * 6 + 54 * 3 - 2 * 10 = 9 + 87 + 6 + 54 * 3 + 2 - 10
= 9 - 8 * 76 - 5 + 43 * 2 * 10, and more 44 equations,
162 = 122 * 32 * 42 * 562 / 72 / 82 / 92 = 12 + 22 * 342 - 52 - 672 + 82 + 92
= 12 + 22 + 32 + 42 * 52 + 62 - 72 - 82 - 92, and more 6 equations,
162 = - 982 / 72 + 62 * 52 - 42 + 32 - 212 = 92 + 82 + 72 + 62 - 52 + 42 + 32 * 22 - 12
= 92 + 82 - 72 + 62 - 52 + 42 * 32 + 22 + 12, and more 2 equations,
162 = 92 * 82 * 72 / 62 / 52 - 42 - 322 / 102 = 982 / 72 * 62 / 52 - 42 - 322 / 102
= 982 / 72 - 62 - 52 + 42 + 32 - 22 + 102, and more 4 equations.
12 + 22 + 32 + ... + 162 = 1496, which consists of different digits.
(162 + 4) = (32 + 4)(42 + 4), (162 - 4) = (42 - 4)(52 - 4).
(1)(2 + 3 + ... + 7)(8 + 9 + ... + 16) = 542,
(1 + 2 + ... + 8)(9 + 10 + ... + 16) = 602,
(1 + 2)(3 + 4 + 5)(6 + 7 + ... + 16) = 662,
(1 + 2)(3 + 4 + ... + 7)(8 + 9 + ... + 16) = 902,
(1 + 2 + ... + 6)(7)(8 + 9 + ... + 16) = 1262,
(13 + 23 + ... + 163) = (1 + 2 + ... + 16)2 = 1362,
(13 + 23 + ... + 153)(163) = 76802,
(products of at most 3 factors)
168 = 4294967296, 4 + 29 + 49 + 6 + 72 + 96 = 42 + 9 + 4 + 96 + 7 + 2 + 96
= 42 + 94 + 9 + 6 + 7 + 2 + 96 = 42 + 94 + 96 + 7 + 2 + 9 + 6 = 162.
1611 = 17592186044416, 1 + 7 + 5 + 92 + 1 + 86 + 04 + 44 + 16 = 162 and more 17 equations.
162 = 256 appears in the decimal expressions of π and e:
π = 3.14159•••256••• (from the 1750th digit),
e = 2.71828•••256••• (from the 1126th digit).
Page of Squares : First Upload September 22, 2003 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan