The smallest squares containing k 38's :
3844 = 622, 138384 = 3722, 3833838724 = 619182,
6383838943876 = 25266262, 3838338138388329 = 619543232.
The first integer which is the sum of a square and a prime in just 4 ways :
12 + 37, 32 + 29, 52 + 13, 62 + 2.
38 is the sum of m squares for m = 3, 4, ..., 24.
The 7th integer which is the sum of 3 distinct squares: 22 + 32 + 52.
The 3rd integer which is the sum of 3 squares in just 2 ways: [1, 1, 6], [2, 3, 5]
The 2nd integer which is the sum of 5 squares in just 2 ways: [1,1,2,4,4], [1,2,2,2,5]
The 3rd integer which is the sum of 6 squares in just 3 ways (see 37).
The 2nd integer which is the sum of 8 squares in just 4 ways (see 36).
382 = 1444, the smallest squares the last three digits of which are equal. There is no square ending in four equal digits.
382 = 24 + 24 + 34 + 34 + 54 + 54.
382 = 1444, where 144 = 122, 4 = 22.
382 = 1444, 12 + 42 + 42 + 42 = 72,
382 = 1444, 1 + 4 + 44 = 72, 382 = 1444, 1 + 44 + 4 = 72.
382 = 2(2! + 6!)
388 = 4347792138496, 4 + 3 + 4 + 7 + 792 + 138 + 496 = 382,
388 = 4347792138496, 4 + 3 + 477 + 92 + 13 + 849 + 6 = 382,
388 = 4347792138496, 43 + 4 + 77 + 921 + 384 + 9 + 6 = 382.
(12 + 22 + 32 + ... + 382) = 19019, which consists of 3 kinds of digits.
382 = (12 + 22 + 32 + ... + 82) + (12 + 22 + 32 + ... + 152).
10k + 218k + 542k + 674k are squares for k = 1,2,3 (382, 8922, 218122).
154k + 222k + 258k + 810k are squares for k = 1,2,3 (382, 8922, 237322).
173k + 245k + 397k + 629k are squares for k = 1,2,3 (382, 8022, 182022).
190k + 314k + 410k + 530k are squares for k = 1,2,3 (382, 7642, 159882).
222k + 354k + 370k + 498k are squares for k = 1,2,3 (382, 7482, 151482).
Komachi Fractions : 382 = 571824/396 = 935712/648 = 1403568/972,
(29/38)2 = 30276/51984.
Komachi equations:
382 = 123 / 4 * 5 / 6 * 7 * 8 + 9 = 12 + 3 + 4 * 5 * 67 + 89
= - 123 + 4 * 56 * 7 + 8 - 9, and more 2 equations,
382 = 9 * 87 + 654 + 3 * 2 + 1 = - 9 * 8 + 76 * 5 * 4 - 3 - 2 + 1,
382 = - 9 * 8 + 76 * 5 * 4 + 3 * 2 - 10 = - 98 + 76 * 5 * 4 + 32 - 10,
382 = 12 * 22 * 32 * 42 + 52 * 62 - 72 - 82 + 92 = 12 + 22 * 32 - 452 - 672 + 892,
382 = 92 + 82 - 72 + 62 * 52 + 42 - 32 + 212 = 92 - 82 - 72 + 62 * 52 + 42 * 32 * 22 * 12
= 92 - 82 - 72 + 62 * 52 + 42 * 32 * 22 / 12.
(382 - 4) = (62 - 4)(72 - 4), (382 + 5) = (42 + 5)(82 + 5),
(382 + 8) = (52 + 8)(62 + 8).
382 + 392 + 402 + ... + 482 = 1432,
382 + 392 + 402 + ... + 962 = 5312,
382 + 392 + 402 + ... + 3492 = 37702,
382 + 392 + 402 + ... + 6862 = 103842,
382 + 392 + 402 + ... + 119182 = 7512282.
(1)(2 + 3 + 4 + 5 + 6)(7 + 8 + ... + 38) = 1202,
(1 + 2 + 3)(4 + 5)(6 + 7 + ... + 38) = 1982,
(1 + 2 + 3)(4 + 5 + ... + 24)(25 + 26 + ... + 38) = 8822.
13 + 23 + ... + 383 = (1 + 2 + ... + 38)2 = 7412.
(13)(23 + 43 + 53)(63 + 73 + ... + 383) = 110882.
382 = 1444 appears in the decimal expressions of π and e:
π = 3.14159•••1444••• (from the 3475th digit),
e = 2.71828•••1444••• (from the 22233rd digit).
Page of Squares : First Upload December 8, 2003 ; Last Revised January 27, 2011
by Yoshio Mimura, Kobe, Japan