730
The smallest squares containing k 730's :
173056 = 4162,
1373073025 = 370552,
1730730730610704 = 416020522.
7302 is the 10th square which is the sum of 4 sixth powers : 16 + 36 + 36 + 96.
Kaprekar : 7306 = 151334226289000000,
and 12 + 52 + 12 + 32 + 32 + 42 + 222 + 62 + 22 + 82 + 92 + 02 + 02 + 02 + 02 + 02 + 02.
(13 + 23 + ... + 2213)(2223 + 2233 + ... + 5783)(5793 + 5803 + ... + 7303) = 8438560361431202.
1 / 730 = 0.001369..., and 1369 = 372.
730k + 3970k + 4190k + 8010k are squares for k = 1,2,3 (1302, 99002, 8065002).
730k + 41902k + 71978k + 77234k are squares for k = 1,2,3 (4382, 1135882, 301195082).
The square root of 730 is 27. 0 1 8 5 12 1 7 2 21..., and 272 = 02 + 12 + 82 + 52 + 122 + 12 + 72 + 22 + 212,
the square root of 730 is 27. 0 18 5 1 2 1 7 2 2 1 2 5 9 2 0 6 1 7 4 6 8 ...,
and 272 = 02 + 182 + 52 + 12 + 22 + 12 + 72 + 22 + 22 + 12 + 22 + 52 + 92 + 22 + 02 + 62 + 12 + 72 + 42 + 62 + 82.
by Yoshio Mimura, Kobe, Japan
731
The smallest squares containing k 731's :
417316 = 6462,
7312473169 = 855132,
273173197314304 = 165279522.
731 = (12 + 22 + 32 + ... + 852) / (12 + 22 + 32 + ... + 92).
7312 = 534361, a zigzag square.
7312 = 30 + 31 + 36 + 37 + 312.
3-by-3 magic squares consisting of different squares with constant 7312:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(7, 426, 594, 486, 441, 322, 546, 398, 279), | (30, 394, 615, 425, 510, 306, 594, 345, 250), |
(34, 153, 714, 441, 574, 102, 582, 426, 119), | (34, 201, 702, 306, 642, 169, 663, 286, 114), |
(34, 306, 663, 471, 498, 254, 558, 439, 174), | (42, 249, 686, 414, 574, 183, 601, 378, 174), |
(54, 393, 614, 439, 474, 342, 582, 394, 201), | (57, 146, 714, 246, 678, 119, 686, 231, 102), |
(78, 254, 681, 474, 537, 146, 551, 426, 222), | (114, 426, 583, 502, 471, 246, 519, 362, 366), |
(153, 306, 646, 366, 601, 198, 614, 282, 279) |
7312 = 534361, 5 + 3 + 4 + 36 + 1 = 72,
7312 = 534361, 5 + 34 + 3 + 6 + 1 = 72,
7312 = 534361, 53 + 4 + 3 + 61 = 112,
7312 = 534361, 5 + 34 + 361 = 202.
7312 + 7322 + 7332 + ... + 7772 = 51702.
Page of Squares : First Upload September 12, 2005 ; Last Revised August 29, 2011by Yoshio Mimura, Kobe, Japan
732
The smallest squares containing k 732's :
677329 = 8232,
732947329 = 270732,
70327327327321 = 83861392.
7322 = 535824, 53 + 5 + 82 + 4 = 122.
7322 is the 7th square which is the sum of 10 sixth powers.
Page of Squares : First Upload September 12, 2005 ; Last Revised September 4, 2006by Yoshio Mimura, Kobe, Japan
733
The smallest squares containing k 733's :
373321 = 6112,
173327338276 = 4163262,
2273307733733089 = 476792172.
7332 = 537289, a square with different digits.
7332 = 537289, 5 + 3 + 72 + 89 = 132.
7332 = 322 + 2912 + 6722 : 2762 + 1922 + 232 = 3372.
Komachi Fraction : 729 / 4835601 = (9 / 733)2.
Kaprekar : 7337 = 113691454465110461077, and
112 + 32 + 62 + 92 + 12 + 42 + 52 + 42 + 42 + 62 + 52 + 12 + 102 + 42 + 62 + 102 + 72 + 72 =733.
3-by-3 magic squares consisting of different squares with constant 7332:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(21, 252, 688, 508, 501, 168, 528, 472, 189), | (24, 192, 707, 437, 564, 168, 588, 427, 96), |
(32, 363, 636, 456, 508, 267, 573, 384, 248), | (60, 267, 680, 405, 580, 192, 608, 360, 195), |
(67, 336, 648, 504, 492, 203, 528, 427, 276), | (99, 328, 648, 472, 468, 309 552, 459, 148), |
(108, 333, 644, 364, 588, 243, 627, 284, 252) |
Page of Squares : First Upload September 12, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan
734
The smallest squares containing k 734's :
73441 = 2712,
3867347344 = 621882,
2734734393017344 = 522946882.
7342 = 538756, 53 + 87 + 56 = 142.
1 / 734 = 0.0013623, 132+62+232 = 734.
7342 = 538756 appears in the decimal expression of π:
π = 3.14159•••538756••• (from the 24873rd digit).
by Yoshio Mimura, Kobe, Japan
735
The smallest squares containing k 735's :
273529 = 5232,
7357350625 = 857752,
7357735073572081 = 857772412.
735 = (12 + 22 + 32 + ... + 492) / (12 + 22 + 32 + 42 + 52).
7352 = 2242 + 4552 + 5322 = 2352 + 5542 + 4222.
(326 / 735)2 = 0.196725438... (Komachic).
Komachi equations:
7352 = 92 * 82 * 72 / 62 * 52 / 42 / 32 * 212 = 982 * 72 * 62 * 52 / 42 * 32 / 212
= 982 / 72 * 62 * 52 / 42 / 32 * 212.
3-by-3 magic squares consisting of different squares with constant 7352:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(5, 190, 710, 242, 670, 181, 694, 235, 58), | (5, 190, 710, 410, 590, 155, 610, 395, 110), |
(5, 274, 682, 410, 565, 230, 610, 382, 149), | (5, 410, 610, 454, 478, 325, 578, 379, 250), |
(7, 224, 700, 476, 532, 175, 560, 455, 140), | (10, 85, 730, 293, 670, 74, 674, 290, 43), |
(10, 85, 730, 470, 562, 59, 565, 466, 62), | (10, 170, 715, 370, 619, 142, 635, 358, 94), |
(10, 170, 715, 506, 517, 130, 533, 494, 110), | (10, 286, 677, 470, 523, 214, 565, 430, 190), |
(10, 370, 635, 470, 485, 290, 565, 410, 230), | (11, 230, 698, 302, 635, 214, 670, 290, 85), |
(26, 118, 725, 155, 710, 110, 718, 149, 50), | (34, 187, 710, 437, 566, 170, 590, 430, 85), |
(36, 345, 648, 423, 540, 264, 600, 360, 225), | (38, 166, 715, 341, 638, 130, 650, 325, 110), |
(50, 347, 646, 485, 470, 290, 550, 446, 197), | ( 50, 485, 550, 514, 370, 373, 523, 410, 314), |
(56, 217, 700, 308, 644, 175, 665, 280, 140), | (60, 324, 657, 495, 468, 276, 540, 465, 180), |
(72, 225, 696, 396, 600, 153, 615, 360, 180), | (85, 290, 670, 430, 565, 190, 590, 370, 235), |
(85, 362, 634, 430, 491, 338, 590, 410, 155), | (106, 250, 683, 283, 650, 194, 670, 235, 190), |
(106, 458, 565, 485, 470, 290, 542, 331, 370), | (107, 326, 650, 370, 590, 235, 626, 293, 250), |
(110, 302, 661, 395, 586, 202, 610, 325, 250), | (110, 395, 610, 475, 506, 242, 550, 358, 331), |
(122, 421, 590, 454, 422, 395, 565, 430, 190), | (135, 384, 612, 480, 513, 216, 540, 360, 345), |
(155, 410, 590, 502, 370, 389, 514, 485, 202), | (166, 370, 613, 410, 565, 230, 587, 290, 334), |
(190, 395, 590, 422, 554, 235, 571, 278, 370) |
7352 = 540225, 5 + 4 + 0 + 2 + 25 = 62,
7352 = 540225, 5 + 4 + 0 + 22 + 5 = 62,
7352 = 540225, 54 + 0 + 2 + 25 = 92,
7352 = 540225, 54 + 0 + 22 + 5 = 92,
7352 = 540225, 5402 + 2252 = 5852.
by Yoshio Mimura, Kobe, Japan
736
The smallest squares containing k 736's :
20736 = 1442,
736796736 = 271442,
298717367367364 = 172834422.
The squares which begin with 736 and end in 736 are
736796736 = 271442, 73634078736 = 2713562, 736411124736 = 8581442,
736775022736 = 8583562, 7361150364736 = 27131442,...
7362± 3 are primes.
7362 = (12 + 7)(32 + 7)(652 + 7) = (112 + 7)(652 + 7) = (42 + 7)(52 + 7)(272 + 7).
7362 = 541696, 5 + 4 + 16 + 96 = 112,
7362 = 541696, 5 + 41 + 69 + 6 = 112.
by Yoshio Mimura, Kobe, Japan
737
The smallest squares containing k 737's :
737881 = 8592,
2173797376 = 466242,
573773707373476 = 239535742.
7372 = 543169, a square with different digits.
Komachi equations:
7372 = 9 - 8 - 7 * 6 + 543210 = 9 - 8 * 7 + 6 + 543210.
10318k + 88440k + 159929k + 284482k are squares for k = 1,2,3 (7372, 3382832, 1667528832).
3-by-3 magic squares consisting of different squares with constant 7372:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(17, 84, 732, 372, 633, 64, 636, 368, 57), | (17, 204, 708, 444, 568, 153, 588, 423, 136), |
(28, 81, 732, 192, 708, 71, 711, 188, 48), | (28, 273, 684, 504, 492, 217, 537, 476, 168), |
(36, 332, 657, 487, 504, 228, 552, 423, 244), | (48, 199 ,708, 249, 672, 172, 692, 228, 111), |
(48, 216, 703, 447, 568, 144, 584, 417, 168), | (48, 456, 577, 496, 447, 312, 543, 368, 336), |
(57, 288, 676, 388, 564, 273, 624, 377, 108), | (108, 217, 696, 316, 648, 153, 657, 276, 188), |
(108, 244, 687, 336, 633, 172, 647, 288, 204), | (108, 431, 588, 512, 468, 249, 519, 372, 368), |
(120, 388, 615, 487, 420, 360, 540, 465, 188) |
7372 = 543169, 5 + 43 + 1 + 6 + 9 = 82,
7372 = 543169, 54 + 31 + 6 + 9 = 102,
7372 = 543169, 53 + 43 + 33 + 163 + 93 = 712.
by Yoshio Mimura, Kobe, Japan
738
The smallest squares containing k 738's :
173889 = 4172,
67380738084 = 2595782,
1380738738673849 = 371582932.
738 = (12 + 22 + 32 + ... + 402) / (12 + 22 + 32 + 42).
7382 = 544644, a square with just 3 kinds of digits.
7382 = (12 + 5)(62 + 5)(472 + 5) = (22 + 2)(42 + 2)(712 + 2).
7382 = 544644, 5 + 4 + 4 + 64 + 4 = 92,
7382 = 544644, 54 + 46 + 44 = 122,
7382 = 544644, 544 + 44 + 644 + 44 = 50282.
Kaprekar : 7387 = 119232467787562584192,
and 112 + 92 + 22 + 32 + 22 + 42 + 62 + 72 + 72 + 82 + 72 + 52 +62 + 22 + 52 + 82 + 42 + 12 + 92 + 22 = 738.
7382 = 34 + 94 + 94 + 274.
Page of Squares : First Upload September 12, 2005 ; Last Revised December 7, 2013by Yoshio Mimura, Kobe, Japan
739
The smallest squares containing k 739's :
7396 = 862,
92739739024 = 3045322,
157397398739344 = 125458122.
7392 = 546121, a zigzag square.
3-by-3 magic squares consisting of different squares with constant 7392:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(1, 138, 726, 294, 666, 127, 678, 289, 54), | (1, 366, 642, 498, 474, 271, 546, 433, 246), |
(6, 303, 674, 399, 566, 258, 622, 366, 159), | (15, 386, 630, 450, 495, 314, 586, 390, 225), |
(33, 134, 726, 474, 561, 82, 566, 462, 111), | (54, 334, 657, 369, 558, 314, 638, 351, 126), |
(54, 447, 586, 478, 426, 369, 561, 406, 258), | (62, 366, 639, 414, 513, 334, 609, 386, 162), |
(63, 246, 694, 314, 639, 198, 666, 278, 159), | (111, 282, 674, 414, 586, 177, 602, 351, 246), |
(114, 271, 678, 426, 582, 161, 593, 366, 246) |
7392 = 546121, 5 + 46 + 12 + 1 = 82,
7392 = 546121, 54 + 6 + 1 + 2 + 1 = 82.
by Yoshio Mimura, Kobe, Japan