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160 - 169

160

The smallest squares containing k 160's :
1600 = 402,
16016004 = 40022,
16005160416025 = 40006452.

A, B, C, A + B, B + C, and C + A are squares for A = 1602, B = 2312, C = 7922.

(122 + 4)(132 + 4) = (1602 + 4).

1602 = 67 x 68 + 69 x 70 + 71 x 72 + 73 x 74 + 75 x 76.

12 + 22 + 32 + ... + 1602 = 1378160, 12 + 32 + 72 + 82 + 12 + 62 = 160.

(1 + 2 + ... + 12)(13 + 14 + ... + 39)(40 + 41 + ... + 160) = 257402,
(1 + 2 + ... + 15)(16 + 17 + ... + 64)(65 + 66 + ... + 160) = 504002,
(1 + 2 + ... + 17)(18 + 19 + ... + 136)(137 + 138 + ... + 160) = 706862,
(1 + 2 + ... + 24)(25 + 26 + ... + 145)(146 + 147 + ... + 160) = 841502,
(1 + 2 + ... + 49)(50 + 51 + ... + 55)(56 + 57 + ... + 160) = 661502,
(1 + 2 + ... + 87)(88 + 89 + ... + 131)(132 + 133 + ... + 160) = 2794442.

1602 = 25600 appears in the decimal expressions of π and e:
  π = 3.14159•••25600••• (from the 20080th digit),
  e = 2.71828•••25600••• (from the 15774th digit)

Page of Squares : First Upload June 28, 2004 ; Last Revised November 1, 2011
by Yoshio Mimura, Kobe, Japan

161

The smallest squares containing k 161's :
14161 = 1192,
147161161 = 121312,
16191618302161 = 40238812.

The squares which begin with 161 and end in 161 are
1610497161 = 401312,   16159240161 = 1271192,   16162291161 = 1271312,
161097074161 = 4013692,   161106707161 = 4013812,...

1 / 161 = 0.006211, 62 + 22 + 112 = 161.

1612 = 25921, 2 + 5 + 92 + 1 = 102,
1612 = 25921, 2 + 59 + 2 + 1 = 82.

1612 = 1! + 6! + 5 x 7!.

A Cubic Polynomial : (X + 442)(X + 572)(X + 1442) = X3 + 1612X2 + 106682X + 3611522.

22k + 161k + 266k + 280k are squares for k = 1,2,3 (272, 4192, 67052).
897k + 2806k + 9890k + 12328k are squares for k = 1,2,3 (1612, 160772, 16922712).

A 3-by-3 magic square consisting of different squares with constant 1612:

1227121442
11121082442
1162962572

(22 - 1)(52 - 1)(192 - 1) = (1612 - 1),
(122 + 5)(132 + 5) = (1612 + 5),
(62 - 5)(292 - 5) = (1612 - 5),
(42 - 5)(62 - 5)(92 - 5) = (1612 - 5).

(1 + 2 + 3)(4 + 5 + 6 + 7 + 8)(9 + 10 + ... + 161) = 15302,
(1 + 2 + 3)(4 + 5 + ... + 14)(15 + 16 + ... + 161) = 27722,
(1 + 2 + 3)(4 + 5 + ... + 66)(67 + 68 + ... + 161) = 119702,
(1 + 2 + 3 + 4 + 5 + 6)(7 + 8 + ... + 42)(43 + ... + 161) = 149942,
(1 + 2 + 3 + 4 + 5 + 6)(7 + 8 + ... + 153)(154 + 155 + ... + 161) = 176402,
(1 + 2 + ... + 8)(9)(10 + 11 + ... + 161) = 20522,
(1 + 2 + ... + 8)(9 + 10 + ... + 144)(145 + 146 + ... + 161) = 312122,
(1 + 2 + ... + 9)(10 + 11 + ... + 66)(67 + 68 + ... + 161) = 324902,
(1 + 2 + ... + 36)(37 + 38 + ... + 110)(111 + 112 + ... + 161) = 1585082,
(1 + 2 + ... + 39)(40 + 41 + ... + 143)(144 + 145 + ... + 161) = 1427402.

1612 = 25921 appears in the decimal expression of e:
  e = 2.71828•••25921••• (from the 15311st digit)

Page of Squares : First Upload June 28, 2004 ; Last Revised February 15, 2011
by Yoshio Mimura, Kobe, Japan

162

The smallest squares containing k 162's :
116281 = 3412,
16216239649 = 1273432,
162041621416249 = 127295572.

1622 = 26244, a square with even digits.

1622 = (12 + 2)(42 + 2)(222 + 2).

1622 = 26244, 2 + 6 + 24 + 4 = 62,
1622 = 26244, 26 + 2 + 4 + 4 = 62.

1622 = 93 + 183 + 273 = 94 + 94 + 94 + 94 = 38 + 38 + 38 + 38.

A+B, A+C, A+D, B+C, B+D, and C+D are squares for (A, B, C, D) = (162, 567, 1282, 4194).

Komachi equations:
1622 = 92 - 82 - 72 + 62 + 542 * 32 - 22 */ 12 = - 92 + 82 + 72 - 62 + 542 * 32 + 22 */ 12.

(122 + 6)(132 + 6) = (1622 + 6),
(22 + 6)(32 + 6)(132 + 6) = (1622 + 6).

(1)(2 + 3 + 4 + 5 + 6 + 7)(8 + 9 + 10)(11 + 12 + 13) = 1622.
(1)(2 + 3 + ... + 25)(26 + 27 + 28) = 1622,
(1)(2 + 3 + 4)(5 + 6 + ... + 76) = 1622,
(1 + 2 + ... + 49)(50 + 51 + ... + 82)(83 + 84 + ... + 162) = 1617002,
(1 + 2 + ... + 49)(50 + 51 + ... + 97)(98 + 99 + ... + 162) = 1911002,
(1 + 2 + ... + 81)(82)(83 + 84 + ... + 162) = 516602.

(13 + 23 + ... + 133)(143 + 153 + ... + 903)(913 + 923 + ... + 1623) = 46762475762.

1622 = 26244 appears in the decimal expressions of π and e:
  π = 3.14159•••26244••• (from the 59833rd digit),
  e = 2.71828•••26244••• (from the 16655th digit)

Page of Squares : First Upload June 28, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

163

The smallest squares containing k 163's :
16384 = 1282,
16301416329 = 1276772,
904163163321636 = 300693062.

1632 = 26569, 26 + 5 + 69 = 102.

1632 = 26569.

1632 = 33 + 193 + 273.

1632 = 30 + 34 + 35 + 38 + 39.

Komachi Fraction : 1632 = 5738904/216.

Komachi equation: 1632 = 93 + 83 + 73 + 63 - 53 * 43 + 323 + 13.

A 3-by-3 magic square consisting of different squares with constant 1632:

1829821292
11121022622
1182812782

(122 + 7)(132 + 7) = (1632 + 7),
(22 + 7)(32 + 7)(122 + 7) = (1632 + 7).

(1 + 2 + ... + 15)(16)(17 + 18 + ... + 163) = 50402,
(1 + 2 + ... + 95)(96 + 97 + ... + 159)(160 + 161 + ... + 163) = 1550402,
(1 + 2 + ... + 162)(163) = 14672.

1632 = 26569 appears in the decimal expression of π:
  π = 3.14159•••26569••• (from the 8464th digit).

Page of Squares : First Upload June 28, 2004 ; Last Revised August 26, 2011
by Yoshio Mimura, Kobe, Japan

164

The smallest squares containing k 164's :
20164 = 1422,
456164164 = 213582,
11641641640225 = 34119852.

(71 / 164)2 = 0.187425639... (Komachic).

The squares which begin with 164 and end in 164 are
16420372164 = 1281422,   16475776164 = 1283582,   164140040164 = 4051422,
164315108164 = 4053582,   164545432164 = 4056422,...

1642 = 26896, 23 + 63 + 83 + 93 + 63 = 412,
1642 = 26896, 26 + 8 + 9 + 6 = 72,
1642 = 26896, 26 + 89 + 6 = 112.

(12 + 22 + 32 + ... + 1642) = 1483790, which consists of different digits.

10k + 164k + 362k + 833k are squares for k = 1,2,3 (372, 9232, 250972).

Komachi equations:
1642 = 9 + 8 + 7 * 6 * 5 * 4 * 32 - 1,
1642 = - 982 - 762 + 52 - 432 + 2102,
1642 = 124 + 34 - 44 - 54 - 64 - 74 + 84 + 94.

(32 - 2)(622 - 2) = (1642 - 2),
(122 + 8)(132 + 8) = (1642 + 8).

(1 + 2 + ... + 11)(12 + 13 + ... + 132)(133 + 134 + ... + 164) = 522722,
(1 + 2 + ... + 17)(18 + 19 + ... + 73)(74 + 75 + ... + 164) = 649742,
(1 + 2 + ... + 31)(32 + 33 + ... + 52)(53 + 54 + ... + 164) = 729122,
(1 + 2 + ... + 50)(51 + 52 + ... + 101)(102 + 103 + ... + 164) = 2034902,
(1 + 2 + ... + 104)(105 + 106 + ... + 129)(130 + 131 + ... + 164) = 2866502.

(12 + 22 + ... + 92)(102 + 112 + ... + 1072)(1082 + 1092 + ... + 1642) = 112318502.

(13 + 23 + ... + 543)(553 + 563 + ... + 993)(1003 + 1013 + ... + 1643) = 882972090002,
(13 + 23 + ... + 1323)(1333 + 1343 + ... + 1643) = 903782882,
(13 + 23 + ... + 1433)(1443 + 1453 + ... + 1643) = 903782882.

1642 = 26896 appears in the decimal expressions of π and e:
  π = 3.14159•••26896••• (from the 80129th digit),
  e = 2.71828•••26896••• (from the 123246th digit)

Page of Squares : First Upload June 28, 2004 ; Last Revised February 15, 2011
by Yoshio Mimura, Kobe, Japan

165

The smallest squares containing k 165's :
165649 = 4072,
516516529 = 227272,
2721655165165225 = 521694852.

1652 = 27225, a square containing 3 kinds of digits.

1652 = 27225, 2 + 7 + 2 + 25 = 62,
1652 = 27225, 2 + 7 + 22 + 5 = 62,
1652 = 27225, 2 + 72 + 2 + 5 = 92,
1652 = 27225, 2 + 722 + 5 = 272,
1652 = 27225, 27 + 2 + 2 + 5 = 62.

(12 + 22 + 32 + ... + 1652) = 1511015, which consists of 3 kinds of digits (the first 7-digit sum).

165k + 7029k + 9273k + 22737k are squares for k = 1,2,3 (1982, 255422, 35915222).

Komachi equations:
1652 = 1 / 2 * 3 * 4 * 567 * 8 + 9 = 9 + 8 * 7 * 6 * 54 * 3 / 2 */ 1,
1652 = 13 * 23 + 33 + 43 + 53 * 63 + 73 + 83 - 93
  = - 93 + 83 + 73 + 63 * 53 + 43 + 33 + 23 */ 13.

Two 3-by-3 magic squares consisting of different squares with constant 1652:

524021602
6421482352
1522612202
     
524021602
10421252282
12821002292

(122 + 9)(132 + 9) = (1652 + 9).

(1 + 2 + 3 + 4 + 5)(6 + 7 + ... + 60) = 1652,
(1 + 2)(3)(4 + 5 + ... + 165) = 3512,
(1 + 2)(3 + 4 + ... + 123)(124 + 125 + ... + 165) = 117812,
(1 + 2 + ... + 5)(6 + 7 + ... + 30)(31 + 32 + ... + 165) = 94502,
(1 + 2 + ... + 20)(21 + 22 + ... + 95)(96 + 97 + ... + 165) = 913502,
(1 + 2 + ... + 24)(25 + 26 + ... + 69)(70 + 71 + ... + 165) = 846002,
(1 + 2 + ... + 26)(27 + 28 + ... + 90)(91 + 92 + ... + 165) = 1123202,
(1 + 2 + ... + 34)(35 + 36 + ... + 84)(85 + 86 + ... + 165) = 1338752,
(1 + 2 + ... + 49)(50 + 51 + ... + 79)(80 + 81 + ... + 165) = 1580252,
(1 + 2 + ... + 50)(51 + 52 + ... + 84)(85 + 86 + ... + 165) = 1721252,
(1 + 2 + ... + 67)(68 + 69 + ... + 102)(103 + 104 + ... + 165) = 2391902,
(13 + 23 + ... + 393)(403 + 413 + ... + 653)(663 + 673 + ... + 1653) = 210810600002.

1652 = 27225 appears in the decimal expression of π:
  π = 3.14159•••27225••• (from the 65583rd digit).

Page of Squares : First Upload July 5, 2004 ; Last Revised February 15, 2011
by Yoshio Mimura, Kobe, Japan

166

The smallest squares containing k 166's :
11664 = 1082,
11669616676 = 1080262,
1166166731664 = 10798922.

The square root of 166 is 12.884..., 122 = 82 + 82 + 42.

Komachi equation: 1662 = 98 + 7 * 654 * 3 * 2 - 10.

1662 = 27556, 2 + 7 + 5 + 5 + 6 = 52.

(32 + 4)(462 + 4) = (1662 + 4),
(12 + 4)(72 + 4)(102 + 4) = (32 + 4)(62 + 4)(72 + 4) = (1662 + 4),
(12 + 4)(22 + 4)(32 + 4)(72 + 4) = (1662 + 4).

(1 + 2 + ... + 15)(16 + 17 + ... + 75)(76 + 77 + ... + 166) = 600602,
(1 + 2 + ... + 21)(22)(23 + 24 + ... + 166) = 83162,
(1 + 2 + ... + 36)(37 + 38 + ... + 148)(149 + 150 + ... + 166) = 1398602,
(1 + 2 + ... + 49)(50 + 51 + ... + 58)(59 + 60 + ... + 166) = 850502,
(1 + 2 + ... + 65)(66 + 67 + ... + 130)(131 + 132 + ... + 166) = 2702702,
(1 + 2 + ... + 128)(129)(130 + 131 + ... + 166) = 763682.

1662 = 27556 appears in the decimal expressions of π and e:
  π = 3.14159•••27556••• (from the 89090th digit),
  e = 2.71828•••27556••• (from the 40235th digit)

Page of Squares : First Upload July 5, 2004 ; Last Revised April 23, 2010
by Yoshio Mimura, Kobe, Japan

167

The smallest squares containing k 167's :
167281 = 4092,
167167317321 = 4088612,
116716735816704 = 108035522.

167 is the sixth prime for which the Legendre Symbol (a/167) = 1 for a = 1, 2, 3, 4.

1672 = 53 + 153 + 293.

1672 = 27889, with a non-decreaing sequence of digits.

1672 = 27889, 2 + 78 + 89 = 132.

167 is the first square which is the sum of a prime and a square in 6 ways :
22 + 163, 42 + 151, 62 + 131, 82 + 103, 102 + 67, 122 + 23.

Two 3-by-3 magic squares consisting of different squares with constant 1672:

329421382
11421022672
1222932662
     
1427821472
9321262582
1382772542

(1 + 2 + ... + 63)(64 + 65 + ... + 112)(113 + 114 + ... + 167) = 2587202.

1672 = 27889 appears in the decimal expression of e:
  e = 2.71828•••27889••• (from the 115409th digit)

Page of Squares : First Upload July 5, 2004 ; Last Revised January 7, 2009
by Yoshio Mimura, Kobe, Japan

168

The smallest squares containing k 168's :
1681 = 412,
1688716836 = 410942,
168168168113089 = 129679672.

1682 = 28224, a square with 3 kinds of even digits.

1682 = 28224, 2 + 8 + 2 + 24 = 62,
1682 = 28224, 2 + 8 + 22 + 4 = 62,
1682 = 28224, 28 + 2 + 2 + 4 = 62.

1682± 5 are primes.

1682 = 223 + 263, the 8th square which is the sum of 2 cubes.

1682 = 18 x 19 x 20 + 20 x 21 x 22 + 22 x 23 x 24.

1682 = (12 + 3)(22 + 3)(32 + 3)(92 + 3) = (22 - 1)(972 - 1) = (32 + 3)(52 + 3)(92 + 3).

30k + 57k + 168k + 474k are squares for k = 1,2,3 (272, 5072, 105572).

Komachi equations:
1682 = 1 - 2 - 3 + 4 + 56 * 7 * 8 * 9 = 12 - 3 * 4 + 56 * 7 * 8 * 9
  = 12 / 3 - 4 + 56 * 7 * 8 * 9 = 12 / 3 / 4 * 56 * 7 * 8 * 9
  = - 1 + 2 + 3 - 4 + 56 * 7 * 8 * 9 = - 12 + 3 * 4 + 56 * 7 * 8 * 9
  = - 12 / 3 + 4 + 56 * 7 * 8 * 9,
1682 = 12 / 22 * 34562 * 72 / 82 / 92 = 12 / 2342 * 562 * 782 * 92.

1682 + 1692 + 1702 + ... + 2172 = 13652,
1682 + 1692 + 1702 + ... + 4662 = 56812,
1682 + 1692 + 1702 + ... + 18562 = 461662,
1682 + 1692 + 1702 + ... + 25592 = 747502,
1682 + 1692 + 1702 + ... + 526082 = 69666382.

(1 + 2 + ... + 8)(9 + 10 + ... + 40) = 1682,
(1 + 2 + ... + 7)(8 + 9 + ... + 56)(57 + 58 + ... + 168) = 235202,
(1 + 2 + ... + 18)(19 + 20 + 21)(22 + 23 + ... + 168) = 119702,
(1 + 2 + ... + 35)(36 + 37 + ... + 111)(112 + 113 + ... + 168) = 1675802,
(1 + 2 + ... + 48)(49 + 50 + ... + 120)(121 + 122 + ... + 168) = 2227682,
(1 + 2 + ... + 84)(85 + 86 + ... + 119)(120 + 121 + ... + 168) = 2998802,
(13 + 23 + ... + 1043)(1053 + 1063 + ... + 1683) = 715478402.

Page of Squares : First Upload July 5, 2004 ; Last Revised January 13, 2014
by Yoshio Mimura, Kobe, Japan

169

the square of 13.

The smallest squares containing k 169's :
169 = 132,
169338169 = 130132,
16957816932169 = 41179872.

The squares which begin with 169 and end in 169 are
169338169 = 130132,   16903380169 = 1300132,   16961676169 = 1302372,
16968449169 = 1302632,   169115870169 = 4112372,...

1 - 16 - 169, a sequence of squares.

1692 = 28561, a zigzag square with diffrent digits.

52k + 91k + 117k + 169k + 247k is squares (262, 3382, 47322, 692902) for k = 1, 2, 3, 4.

a reversible square (961 = 312).

169 = 169, 169 = 132, 16 = 42, 9 = 32.

169 is the sum of n nonzero squares (n = 1,2,...,155).

169 -- 1! + 6! + 9! = 363601 -- 3! + 6! + 3! + 6! + 0! + 1! = 1454 -- 1! + 4! + 5! + 4! = 169.

Komachi equations:
1692 = 1 + 2 * 34 * 5 / 6 * 7 * 8 * 9,
1692 = 94 * 84 * 74 * 654 / 44 / 34 / 2104.

Two 3-by-3 magic squares consisting of different squares with constant 1692:

1223921642
5621562332
1592522242
     
1626321562
10821242392
1292962522

(42 - 5)(512 - 5) = (1692 - 5),
(42 - 5)(72 - 5)(82 - 5) = (1692 - 5).

1692 = 1192 + 1202 + 1212 + ... + 1202.

(1 + 2 + ... + 20)(21 + 22 + ... + 55)(56 + 57 + ... + 169) = 598502,
(1 + 2 + ... + 24)(25 + 26 + ... + 120)(121 + 122 + ... + 169) = 1218002,
(1 + 2 + ... + 27)(28 + 29 + ... + 43)(44 + 45 + ... + 169) = 536762,
(1 + 2 + ... + 72)(73)(74 + 75 + ... + 169) = 473042,
(1 + 2 + ... + 84)(85 + 86 + ... + 119)(120 + ... + 169 ) = 3034502,
(1 + 2 + ... + 92)(93 + 94 + ... + 138)(139 + ... + 169 ) = 3294062.

1692 = 28561 appears in the decimal expressions of π and e:
  π = 3.14159•••28561••• (from the 11115th digit),
  e = 2.71828•••28561••• (from the 30224th digit)

Page of Squares : First Upload July 5, 2004 ; Last Revised April 23, 2010
by Yoshio Mimura, Kobe, Japan