The smallest squares containing k 37's :
3721 = 612, 2637376 = 16242, 33713733769 = 1836132,
3780376373761 = 19443192, 3737375248373776 = 611340762.
The 4th integer which is the sum of 4 squares in just 2 ways: [1,2,4,4], [2,2,2,5].
The 1st integer which is the sum of 7 squares in just 4 ways (see 36).
372 = 1369, the sequence of its digits is increasing.
372 = 1369, 1 = 12, 36 = 62, 9 = 32.
The sum of consecutive odd primes : 3 + 5 + 7 + 11 + 13 + ... + 107 = 372.
372 = 43 + 43 + 83 + 93.
Loop of length 8 by the function f(N) = ... + c2 + b2 + a2 for N = ... + 102c + 10b + a:
37 -- 58 -- 89 -- 145 -- 42 -- 20 -- 4 -- 16 -- 37
Loop of length 35 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
37 - 1369 - 4930 - 3301 - 1090 - 8200 - 6724 - 5065 - 6725 - 5114 - 2797 - 10138 - 1446 - 2312 - 673 - 5365 - 7034 - 6056 - 6736 - 5785 - 10474 - 5493 - 11565 - 4451 - 4537 - 3394 - 9925 - 10426 - 693 - 8685 - 14621 - 2558 - 3989 - 9442 - 10600 - 37
(Note f(37) = 372 = 1369, f(1369) = 132 + 692 = 4930, etc. See 41)
10k + 164k + 362k + 833k are squares for k = 1,2,3 (372, 9232, 250972).
Komachi Fractions : 372 = 628371/459, (14/37)2 = 10584/73926.
Komachi equations:
372 = 1 + 2 * 3 * 4 * 56 + 7 + 8 + 9 = 1234 + 56 + 7 + 8 * 9
= - 1 + 2 + 345 * 6 - 78 * 9,
372 = 9 + 87 - 6 - 5 + 4 * 321 = 987 + 65 - 4 + 321
= - 9 + 8 + 76 * 54 / 3 + 2 * 1, and more 2 equations,
372 = 98 - 7 + 6 * 5 * 43 - 2 - 10 = 9 * 8 * 7 / 6 + 5 + 4 * 32 * 10
= 98 / 7 * 6 + 5 + 4 * 32 * 10, and more 3 equations,
372 = 12 * 22 + 342 + 562 / 72 + 82 + 92 = 12 - 22 * 342 + 52 - 62 + 782 - 92
= 12 - 22 - 32 - 42 * 52 + 62 * 72 - 82 + 92 = 122 + 32 * 42 * 52 * 62 * 72 / 82 / 92,
372 = 92 + 82 + 72 * 62 * 52 / 42 / 32 * 22 - 12 = 92 + 82 + 72 / 62 * 52 * 42 * 32 / 22 - 12
= 92 - 82 + 72 * 62 - 52 * 42 - 32 - 22 + 12,
372 = - 93 + 83 + 73 + 63 - 53 - 43 + 33 * 23 + 103.
(372 - 5) = (62 - 5)(72 - 5) = (32 - 5)(42 - 5)(62 - 5),
(372 + 7) = (52 + 7)(62 + 7).
(12 + 22 + 32 + ... + 372) = 17575, which consists of odd digits (the first 5-digit).
(1)(2 + 3 + 4 + 5 + 6 + 7)(8 + 9 + ... + 37) = 1352,
(1 + 2)(3 + 4 + 5 + 6 + 7)(8 + 9 + ... + 37) = 2252,
(1 + 2)(3 + 4 + ... + 12)(13 + 14 + ... + 37) = 3752,
(1 + 2)(3 + 4 + ... + 32)(33 + 34 + ... + 37) = 5252,
(1 + 2 + 3 + 4)(5 + 6 + 7 + 8 + 9 + 10)(11 + 12 + ... + 37) = 5402,
(1 + 2 + 3 + 4 + 5 + 6)(7)(8 + 9 + ... + 37) = 3152,
(1 + 2 + 3 + 4 + 5 + 6)(7 + 8 + ... + 32)(33 + 34 + ... + 37) = 13652,
(1 + 2 + ... + 11)(12 + 13 + ... + 32)(33 + 34 + ... + 37) = 23102,
(1 + 2 + ... + 20)(21 + 22 + ... + 30)(31 + 32 + ... + 37) = 35702,
(1 + 2 + ... + 21)(22 + 23 + ... + 28)(29 + 30 + ... + 37) = 34652,
(1 + 2 + ... + 21)(22 + 23 + ... + 32)(33 + 34 + ... + 37) = 34652,
(1 + 2 + ... + 26)(27)(28 + 29 + ... + 37) = 17552,
(1 + 2 + ... + 27)(28 + 29 + ... + 32)(33 + 34 + ... + 37) = 31502,
(1 + 2 + ... + 9)(10)(11 + 12 + ... + 37) = 5402.
13 + 23 + ... + 373 = (1 + 2 + ... + 37)2 = 7032.
375 = 69343957, 69 + 343 + 957 = 372,
378 = 3512479453921,
3 + 5 + 1 + 2 + 4 + 794 + 539 + 21 = 3 + 5 + 1 + 24 + 794 + 539 + 2 + 1
= 3 + 5 + 12 + 4 + 7 + 945 + 392 + 1 = 3 + 5 + 1247 + 9 + 4 + 5 + 3 + 92 + 1
= 3 + 5 + 1247 + 9 + 45 + 39 + 21 = 3 + 5 + 1247 + 94 + 5 + 3 + 9 + 2 + 1
= 3 + 51 + 247 + 94 + 53 + 921 = 35 + 1247 + 9 + 45 + 3 + 9 + 21
= 351 + 2 + 4 + 7 + 945 + 39 + 21 = 351 + 2 + 4 + 79 + 4 + 5 + 3 + 921
= 351 + 24 + 7 + 9 + 4 + 53 + 921 = 351 + 24 + 7 + 945 + 39 + 2 + 1 = 372,
379 = 129961739795077,
1 + 29 + 9 + 61 + 739 + 7 + 9 + 507 + 7 = 1 + 299 + 6 + 17 + 3 + 9 + 7 + 950 + 77
= 1 + 299 + 6 + 17 + 3 + 979 + 50 + 7 + 7 = 1 + 299 + 61 + 7 + 3 + 979 + 5 + 07 + 7
= 12 + 9 + 9 + 6 + 1 + 739 + 79 + 507 + 7 = 12 + 99 + 61 + 73 + 97 + 950 + 77
= 12 + 996 + 173 + 9 + 7 + 95 + 077 = 12 + 996 + 173 + 97 + 9 + 5 + 077
= 129 + 96 + 17 + 3 + 97 + 950 + 77 = 129 + 961 + 7 + 3 + 97 + 95 + 077
= 129 + 961 + 73 + 97 + 95 + 07 + 7 = 1299 + 6 + 17 + 3 + 9 + 7 + 9 + 5 + 07 + 7 = 372.
372 = 1369 appears in the decimal expressions of π and e:
π = 3.14159•••1369••• (from the 3466th digit),
e = 2.71828•••1369••• (from the 16299th digit).
Page of Squares : First Upload December 1, 2003 ; Last Revised January 27, 2011
by Yoshio Mimura, Kobe, Japan