The smallest squares containing k 15's :
1156 = 342, 1515361 = 12312, 1515155625 = 389252,
1527157151524 = 12357822, 1515158661151521 = 389250392.
152 = 225, 2 + 2 + 5 = 32.
152 = 63 + 9.
152± 2 are primes (the third case).
152 is the second square which is the sum of 3 squares in just 3 ways:
22 + 52 + 142 = 22 + 102 + 112 = 52 + 102 + 102.
152 is the sum of 3 distinct cubes: 13 + 23 + 63.
152 is the sum of 5 consecutive cubes: 13 + 23 + 33 + 43 + 53.
152 is the first square which is the sum of 5 distinct squares in 2 ways:
22 + 42 + 52 + 62 + 122 = 32 + 42 + 62 + 82 + 102.
152 is the 2nd square which is the sum of 7 distinct squares: 12 + 22 + 32 + 42 + 52 + 72 + 112.
Komachi Fractions: 152 = 196425/873 = 392850/1746,
(11/15)2 = 20691/38475 = 41382/76950.
Komachi equations:
152 = 1 - 2 + 3 - 4 + 5 * 6 * 7 + 8 + 9 = 1 * 2 + 3 * 4 + 5 * 6 * 7 - 8 + 9
= 12 * 3 - 4 + 5 * 6 * 7 - 8 - 9, and more 54 equations,
152 = 9 + 8 + 7 * 6 * 5 + 4 - 3 - 2 - 1 = 9 + 8 + 7 * 6 * 5 + 4 - 3 * 2 * 1
= 9 + 8 + 7 * 6 * 5 - 4 + 3 - 2 + 1, and more 47 equations,
152 = 9 - 8 + 7 * 6 * 5 + 4 * 3 * 2 - 10 = 9 - 8 + 7 * 6 / 5 * 4 / 3 * 2 * 10
= 9 + 87 + 6 * 5 * 4 - 3 + 2 + 10, and more 46 equations,
152 = 12 * 22 + 32 * 42 - 52 + 62 + 72 - 82 + 92 = 12 * 22 * 32 - 42 - 52 + 62 + 72 + 82 + 92
= - 12 - 22 - 32 - 42 + 52 + 62 + 72 + 82 + 92, and more 2 equations,
152 = 982 - 762 - 52 * 42 * 32 - 22 + 12 = 982 / 72 * 62 * 52 / 42 * 32 / 212
= 92 + 82 + 72 + 62 + 52 - 42 - 32 - 22 - 12, and more 11 equations,
152 = 982 / 72 - 62 + 52 + 42 * 32 - 22 - 102,
152 = 123 / 33 * 43 / 563 * 73 - 83 + 93.
152 = (1 + 2)(3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12),
152 = (1 + 2 + 3 + 4 + 5)2 = 13 + 23 + 33 + 43 + 53.
152 + 3 = (32 + 3)(42 + 3), (152 - 5) = (42 - 5)(52 - 5),
152 + 9 = (22 + 9)(32 + 9).
152 + 162 + 172 + ... + 1112 = 6792,
152 + 162 + 172 + ... + 3262 = 34062.
12 + 22 + 32 + ... + 152 = 1240, which consists of different digits.
(1 + 2 + ... + 8)(9 + 10 + ... + 12)(13 + 14 + 15) = 2522,
(12 + 22 + ... + 72)(82 + 92 + ... + 142)(152) = 52502,
(13 + 23 + ... + 153) = (1 + 2 + ... + 15)2 = 1202,
(products of at most 3 factors)
(14 + 24 + 34)(44)(54)(64)(74)(84)(94)(104)(114)(124 + 134 + 144 + 154) = 1629643915468800002.
158 = 2562890625, 2 + 5 + 62 + 89 + 062 + 5 = 2 + 56 + 2 + 8 + 90 + 62 + 5 = 152,
159 = 38443359375, 3 + 8 + 4 + 4 + 3 + 35 + 93 + 75 = 152 and more 10 equations.
1510 = 576650390625, 5 + 76 + 65 + 03 + 9 + 062 + 5 = 57 + 6 + 6 + 50 + 3 + 90 + 6 + 2 + 5
= 57 + 6 + 6 + 50 + 39 + 062 + 5 = 57 + 66 + 50 + 39 + 06 + 2 + 5 = 152.
1511 = 8649755859375, 8 + 6 + 4 + 9 + 7 + 5 + 5 + 8 + 5 + 93 + 75 = 152 and more 15 equations.
1512 = 129746337890625, 1 + 2 + 9 + 7 + 4 + 6 + 3 + 37 + 89 + 062 + 5 = 152 and more 34 equations.
1513 = 1946195068359375, 1 + 9 + 4 + 6 + 1 + 9 + 5 + 06 + 8 + 3 + 5 + 93 + 75 = 152 and more 42 equations.
152 = 225 appears in the decimal expressions of π and e:
π = 3.14159•••225••• (from the 1070th digit), 225 is the 10th 3-digit square in the expr. of π.
e = 2.71828•••225••• (from the 1247th digit.).
Page of Squares : First Upload September 15, 2003 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan