本文へスキップ
Number Buffet
ja

最初の 1,000 個の不足数

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99, 101, 103, 105, 106, 107, 109, 110, 111, 113, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 127, 128, 129, 130, 131, 133, 134, 135, 136, 137, 139, 141, 142, 143, 145, 146, 147, 148, 149, 151, 152, 153, 154, 155, 157, 158, 159, 161, 163, 164, 165, 166, 167, 169, 170, 171, 172, 173, 175, 177, 178, 179, 181, 182, 183, 184, 185, 187, 188, 189, 190, 191, 193, 194, 195, 197, 199, 201, 202, 203, 205, 206, 207, 209, 211, 212, 213, 214, 215, 217, 218, 219, 221, 223, 225, 226, 227, 229, 230, 231, 232, 233, 235, 236, 237, 238, 239, 241, 242, 243, 244, 245, 247, 248, 249, 250, 251, 253, 254, 255, 256, 257, 259, 261, 262, 263, 265, 266, 267, 268, 269, 271, 273, 274, 275, 277, 278, 279, 281, 283, 284, 285, 286, 287, 289, 290, 291, 292, 293, 295, 296, 297, 298, 299, 301, 302, 303, 305, 307, 309, 310, 311, 313, 314, 315, 316, 317, 319, 321, 322, 323, 325, 326, 327, 328, 329, 331, 332, 333, 334, 335, 337, 338, 339, 341, 343, 344, 345, 346, 347, 349, 351, 353, 355, 356, 357, 358, 359, 361, 362, 363, 365, 367, 369, 370, 371, 373, 374, 375, 376, 377, 379, 381, 382, 383, 385, 386, 387, 388, 389, 391, 393, 394, 395, 397, 398, 399, 401, 403, 404, 405, 406, 407, 409, 410, 411, 412, 413, 415, 417, 418, 419, 421, 422, 423, 424, 425, 427, 428, 429, 430, 431, 433, 434, 435, 436, 437, 439, 441, 442, 443, 445, 446, 447, 449, 451, 452, 453, 454, 455, 457, 458, 459, 461, 463, 465, 466, 467, 469, 470, 471, 472, 473, 475, 477, 478, 479, 481, 482, 483, 484, 485, 487, 488, 489, 491, 493, 494, 495, 497, 499, 501, 502, 503, 505, 506, 507, 508, 509, 511, 512, 513, 514, 515, 517, 518, 519, 521, 523, 524, 525, 526, 527, 529, 530, 531, 533, 535, 536, 537, 538, 539, 541, 542, 543, 545, 547, 548, 549, 551, 553, 554, 555, 556, 557, 559, 561, 562, 563, 565, 566, 567, 568, 569, 571, 573, 574, 575, 577, 578, 579, 581, 583, 584, 585, 586, 587, 589, 590, 591, 592, 593, 595, 596, 597, 598, 599, 601, 602, 603, 604, 605, 607, 609, 610, 611, 613, 614, 615, 617, 619, 621, 622, 623, 625, 626, 627, 628, 629, 631, 632, 633, 634, 635, 637, 638, 639, 641, 643, 645, 646, 647, 649, 651, 652, 653, 655, 656, 657, 658, 659, 661, 662, 663, 664, 665, 667, 668, 669, 670, 671, 673, 674, 675, 676, 677, 679, 681, 682, 683, 685, 686, 687, 688, 689, 691, 692, 693, 694, 695, 697, 698, 699, 701, 703, 705, 706, 707, 709, 710, 711, 712, 713, 715, 716, 717, 718, 719, 721, 722, 723, 724, 725, 727, 729, 730, 731, 733, 734, 735, 737, 739, 741, 742, 743, 745, 746, 747, 749, 751, 752, 753, 754, 755, 757, 758, 759, 761, 763, 764, 765, 766, 767, 769, 771, 772, 773, 775, 776, 777, 778, 779, 781, 782, 783, 785, 787, 788, 789, 790, 791, 793, 794, 795, 796, 797, 799, 801, 802, 803, 805, 806, 807, 808, 809, 811, 813, 814, 815, 817, 818, 819, 821, 823, 824, 825, 826, 827, 829, 830, 831, 833, 835, 837, 838, 839, 841, 842, 843, 844, 845, 847, 848, 849, 850, 851, 853, 854, 855, 856, 857, 859, 861, 862, 863, 865, 866, 867, 869, 871, 872, 873, 874, 875, 877, 878, 879, 881, 883, 884, 885, 886, 887, 889, 890, 891, 892, 893, 895, 897, 898, 899, 901, 902, 903, 904, 905, 907, 908, 909, 911, 913, 914, 915, 916, 917, 919, 921, 922, 923, 925, 926, 927, 929, 931, 932, 933, 934, 935, 937, 938, 939, 941, 943, 944, 946, 947, 949, 950, 951, 953, 955, 956, 957, 958, 959, 961, 962, 963, 964, 965, 967, 969, 970, 971, 973, 974, 975, 976, 977, 979, 981, 982, 983, 985, 986, 987, 988, 989, 991, 993, 994, 995, 997, 998, 999, 1,001, 1,003, 1,004, 1,005, 1,006, 1,007, 1,009, 1,010, 1,011, 1,012, 1,013, 1,015, 1,016, 1,017, 1,018, 1,019, 1,021, 1,022, 1,023, 1,024, 1,025, 1,027, 1,028, 1,029, 1,030, 1,031, 1,033, 1,034, 1,035, 1,037, 1,039, 1,041, 1,042, 1,043, 1,045, 1,046, 1,047, 1,048, 1,049, 1,051, 1,052, 1,053, 1,054, 1,055, 1,057, 1,058, 1,059, 1,061, 1,063, 1,065, 1,066, 1,067, 1,069, 1,070, 1,071, 1,072, 1,073, 1,075, 1,076, 1,077, 1,078, 1,079, 1,081, 1,082, 1,083, 1,084, 1,085, 1,087, 1,089, 1,090, 1,091, 1,093, 1,094, 1,095, 1,096, 1,097, 1,099, 1,101, 1,102, 1,103, 1,105, 1,106, 1,107, 1,108, 1,109, 1,111, 1,112, 1,113, 1,114, 1,115, 1,117, 1,118, 1,119, 1,121, 1,123, 1,124, 1,125, 1,126, 1,127, 1,129, 1,130, 1,131, 1,132, 1,133, 1,135, 1,136, 1,137, 1,138, 1,139, 1,141, 1,142, 1,143, 1,145, 1,147, 1,149, 1,150, 1,151, 1,153, 1,154, 1,155, 1,156, 1,157, 1,159, 1,161, 1,162, 1,163, 1,165, 1,166, 1,167, 1,168, 1,169, 1,171, 1,172, 1,173, 1,174, 1,175, 1,177, 1,178, 1,179, 1,181, 1,183, 1,185, 1,186, 1,187, 1,189, 1,191, 1,192, 1,193, 1,195, 1,196, 1,197, 1,198, 1,199, 1,201, 1,202, 1,203, 1,205, 1,207, 1,208, 1,209, 1,210, 1,211, 1,213, 1,214, 1,215, 1,217, 1,219, 1,221, 1,222, 1,223, 1,225, 1,226, 1,227, 1,228, 1,229, 1,231, 1,233, 1,234, 1,235, 1,237, 1,238, 1,239, 1,241, 1,243, 1,244, 1,245, 1,246, 1,247, 1,249, 1,250, 1,251, 1,252, 1,253, 1,255, 1,256, 1,257, 1,258, 1,259, 1,261, 1,262, 1,263, 1,264, 1,265, 1,267, 1,268, 1,269, 1,270, 1,271, 1,273, 1,274, 1,275, 1,276, 1,277, 1,279, 1,281, 1,282, 1,283, 1,285, 1,286, 1,287, 1,289, 1,291, 1,292, 1,293, 1,294, 1,295, 1,297, 1,298, 1,299, 1,301, 1,303, 1,304, 1,305, 1,306, 1,307, 1,309, 1,310, 1,311, 1,313, 1,315, 1,317, 1,318, 1,319, 1,321, 1,322, 1,323, 1,324, 1,325, 1,327

設定

クイックプリセット

Used in count mode. Terms are produced in ascending order from 1, which is deficient because it has no proper divisors at all.

Used in range mode. Inclusive.

Used in range mode. Inclusive, up to 4,000,000.

Appends 2n − σ(n), the amount by which the divisors fall short. A prime p falls short by p − 1; a power of two by exactly 1.

見た目を微調整

まず画像の横にあるプリセットを選んでください。ここで細かく調整します。

Frame

A border drawn inside the edge of the image.

詳細設定

結果

1,000 件の値

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99, 101, 103, 105, 106, 107, 109, 110, 111, 113, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 127, 128, 129, 130, 131, 133, 134, 135, 136, 137, 139, 141, 142, 143, 145, 146, 147, 148, 149, 151, 152, 153, 154, 155, 157, 158, 159, 161, 163, 164, 165, 166, 167, 169, 170, 171, 172, 173, 175, 177, 178, 179, 181, 182, 183, 184, 185, 187, 188, 189, 190, 191, 193, 194, 195, 197, 199, 201, 202, 203, 205, 206, 207, 209, 211, 212, 213, 214, 215, 217, 218, 219, 221, 223, 225, 226, 227, 229, 230, 231, 232, 233, 235, 236, 237, 238, 239, 241, 242, 243, 244, 245, 247, 248, 249, 250, 251, 253, 254, 255, 256, 257, 259, 261, 262, 263, 265, 266, 267, 268, 269, 271, 273, 274, 275, 277, 278, 279, 281, 283, 284, 285, 286, 287, 289, 290, 291, 292, 293, 295, 296, 297, 298, 299, 301, 302, 303, 305, 307, 309, 310, 311, 313, 314, 315, 316, 317, 319, 321, 322, 323, 325, 326, 327, 328, 329, 331, 332, 333, 334, 335, 337, 338, 339, 341, 343, 344, 345, 346, 347, 349, 351, 353, 355, 356, 357, 358, 359, 361, 362, 363, 365, 367, 369, 370, 371, 373, 374, 375, 376, 377, 379, 381, 382, 383, 385, 386, 387, 388, 389, 391, 393, 394, 395, 397, 398, 399, 401, 403, 404, 405, 406, 407, 409, 410, 411, 412, 413, 415, 417, 418, 419, 421, 422, 423, 424, 425, 427, 428, 429, 430, 431, 433, 434, 435, 436, 437, 439, 441, 442, 443, 445, 446, 447, 449, 451, 452, 453, 454, 455, 457, 458, 459, 461, 463, 465, 466, 467, 469, 470, 471, 472, 473, 475, 477, 478, 479, 481, 482, 483, 484, 485, 487, 488, 489, 491, 493, 494, 495, 497, 499, 501, 502, 503, 505, 506, 507, 508, 509, 511, 512, 513, 514, 515, 517, 518, 519, 521, 523, 524, 525, 526, 527, 529, 530, 531, 533, 535, 536, 537, 538, 539, 541, 542, 543, 545, 547, 548, 549, 551, 553, 554, 555, 556, 557, 559, 561, 562, 563, 565, 566, 567, 568, 569, 571, 573, 574, 575, 577, 578, 579, 581, 583, 584, 585, 586, 587, 589, 590, 591, 592, 593, 595, 596, 597, 598, 599, 601, 602, 603, 604, 605, 607, 609, 610, 611, 613, 614, 615, 617, 619, 621, 622, 623, 625, 626, 627, 628, 629, 631, 632, 633, 634, 635, 637, 638, 639, 641, 643, 645, 646, 647, 649, 651, 652, 653, 655, 656, 657, 658, 659, 661, 662, 663, 664, 665, 667, 668, 669, 670, 671, 673, 674, 675, 676, 677, 679, 681, 682, 683, 685, 686, 687, 688, 689, 691, 692, 693, 694, 695, 697, 698, 699, 701, 703, 705, 706, 707, 709, 710, 711, 712, 713, 715, 716, 717, 718, 719, 721, 722, 723, 724, 725, 727, 729, 730, 731, 733, 734, 735, 737, 739, 741, 742, 743, 745, 746, 747, 749, 751, 752, 753, 754, 755, 757, 758, 759, 761, 763, 764, 765, 766, 767, 769, 771, 772, 773, 775, 776, 777, 778, 779, 781, 782, 783, 785, 787, 788, 789, 790, 791, 793, 794, 795, 796, 797, 799, 801, 802, 803, 805, 806, 807, 808, 809, 811, 813, 814, 815, 817, 818, 819, 821, 823, 824, 825, 826, 827, 829, 830, 831, 833, 835, 837, 838, 839, 841, 842, 843, 844, 845, 847, 848, 849, 850, 851, 853, 854, 855, 856, 857, 859, 861, 862, 863, 865, 866, 867, 869, 871, 872, 873, 874, 875, 877, 878, 879, 881, 883, 884, 885, 886, 887, 889, 890, 891, 892, 893, 895, 897, 898, 899, 901, 902, 903, 904, 905, 907, 908, 909, 911, 913, 914, 915, 916, 917, 919, 921, 922, 923, 925, 926, 927, 929, 931, 932, 933, 934, 935, 937, 938, 939, 941, 943, 944, 946, 947, 949, 950, 951, 953, 955, 956, 957, 958, 959, 961, 962, 963, 964, 965, 967, 969, 970, 971, 973, 974, 975, 976, 977, 979, 981, 982, 983, 985, 986, 987, 988, 989, 991, 993, 994, 995, 997, 998, 999, 1,001, 1,003, 1,004, 1,005, 1,006, 1,007, 1,009, 1,010, 1,011, 1,012, 1,013, 1,015, 1,016, 1,017, 1,018, 1,019, 1,021, 1,022, 1,023, 1,024, 1,025, 1,027, 1,028, 1,029, 1,030, 1,031, 1,033, 1,034, 1,035, 1,037, 1,039, 1,041, 1,042, 1,043, 1,045, 1,046, 1,047, 1,048, 1,049, 1,051, 1,052, 1,053, 1,054, 1,055, 1,057, 1,058, 1,059, 1,061, 1,063, 1,065, 1,066, 1,067, 1,069, 1,070, 1,071, 1,072, 1,073, 1,075, 1,076, 1,077, 1,078, 1,079, 1,081, 1,082, 1,083, 1,084, 1,085, 1,087, 1,089, 1,090, 1,091, 1,093, 1,094, 1,095, 1,096, 1,097, 1,099, 1,101, 1,102, 1,103, 1,105, 1,106, 1,107, 1,108, 1,109, 1,111, 1,112, 1,113, 1,114, 1,115, 1,117, 1,118, 1,119, 1,121, 1,123, 1,124, 1,125, 1,126, 1,127, 1,129, 1,130, 1,131, 1,132, 1,133, 1,135, 1,136, 1,137, 1,138, 1,139, 1,141, 1,142, 1,143, 1,145, 1,147, 1,149, 1,150, 1,151, 1,153, 1,154, 1,155, 1,156, 1,157, 1,159, 1,161, 1,162, 1,163, 1,165, 1,166, 1,167, 1,168, 1,169, 1,171, 1,172, 1,173, 1,174, 1,175, 1,177, 1,178, 1,179, 1,181, 1,183, 1,185, 1,186, 1,187, 1,189, 1,191, 1,192, 1,193, 1,195, 1,196, 1,197, 1,198, 1,199, 1,201, 1,202, 1,203, 1,205, 1,207, 1,208, 1,209, 1,210, 1,211, 1,213, 1,214, 1,215, 1,217, 1,219, 1,221, 1,222, 1,223, 1,225, 1,226, 1,227, 1,228, 1,229, 1,231, 1,233, 1,234, 1,235, 1,237, 1,238, 1,239, 1,241, 1,243, 1,244, 1,245, 1,246, 1,247, 1,249, 1,250, 1,251, 1,252, 1,253, 1,255, 1,256, 1,257, 1,258, 1,259, 1,261, 1,262, 1,263, 1,264, 1,265, 1,267, 1,268, 1,269, 1,270, 1,271, 1,273, 1,274, 1,275, 1,276, 1,277, 1,279, 1,281, 1,282, 1,283, 1,285, 1,286, 1,287, 1,289, 1,291, 1,292, 1,293, 1,294, 1,295, 1,297, 1,298, 1,299, 1,301, 1,303, 1,304, 1,305, 1,306, 1,307, 1,309, 1,310, 1,311, 1,313, 1,315, 1,317, 1,318, 1,319, 1,321, 1,322, 1,323, 1,324, 1,325, 1,327

画像を作成

これらの数字を装飾して画像としてダウンロードするには JavaScript を有効にしてください。値そのものは上に一覧表示されています。

Text on the image

Drag a line straight onto the picture to place it — once placed, it stays exactly where you put it. Everything here is drawn into the download.

最初の 1,000 個の不足数は何ですか？

最初の 1,000 個の不足数は次のとおりです。

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99, 101, 103, 105, 106, 107, 109, 110, 111, 113, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 127, 128, 129, 130, 131, 133, 134, 135, 136, 137, 139, 141, 142, 143, 145, 146, 147, 148, 149, 151, 152, 153, 154, 155, 157, 158, 159, 161, 163, 164, 165, 166, 167, 169, 170, 171, 172, 173, 175, 177, 178, 179, 181, 182, 183, 184, 185, 187, 188, 189, 190, 191, 193, 194, 195, 197, 199, 201, 202, 203, 205, 206, 207, 209, 211, 212, 213, 214, 215, 217, 218, 219, 221, 223, 225, 226, 227, 229, 230, 231, 232, 233, 235, 236, 237, 238, 239, 241, 242, 243, 244, 245, 247, 248, 249, 250, 251, 253, 254, 255, 256, 257, 259, 261, 262, 263, 265, 266, 267, 268, 269, 271, 273, 274, 275, 277, 278, 279, 281, 283, 284, 285, 286, 287, 289, 290, 291, 292, 293, 295, 296, 297, 298, 299, 301, 302, 303, 305, 307, 309, 310, 311, 313, 314, 315, 316, 317, 319, 321, 322, 323, 325, 326, 327, 328, 329, 331, 332, 333, 334, 335, 337, 338, 339, 341, 343, 344, 345, 346, 347, 349, 351, 353, 355, 356, 357, 358, 359, 361, 362, 363, 365, 367, 369, 370, 371, 373, 374, 375, 376, 377, 379, 381, 382, 383, 385, 386, 387, 388, 389, 391, 393, 394, 395, 397, 398, 399, 401, 403, 404, 405, 406, 407, 409, 410, 411, 412, 413, 415, 417, 418, 419, 421, 422, 423, 424, 425, 427, 428, 429, 430, 431, 433, 434, 435, 436, 437, 439, 441, 442, 443, 445, 446, 447, 449, 451, 452, 453, 454, 455, 457, 458, 459, 461, 463, 465, 466, 467, 469, 470, 471, 472, 473, 475, 477, 478, 479, 481, 482, 483, 484, 485, 487, 488, 489, 491, 493, 494, 495, 497, 499, 501, 502, 503, 505, 506, 507, 508, 509, 511, 512, 513, 514, 515, 517, 518, 519, 521, 523, 524, 525, 526, 527, 529, 530, 531, 533, 535, 536, 537, 538, 539, 541, 542, 543, 545, 547, 548, 549, 551, 553, 554, 555, 556, 557, 559, 561, 562, 563, 565, 566, 567, 568, 569, 571, 573, 574, 575, 577, 578, 579, 581, 583, 584, 585, 586, 587, 589, 590, 591, 592, 593, 595, 596, 597, 598, 599, 601, 602, 603, 604, 605, 607, 609, 610, 611, 613, 614, 615, 617, 619, 621, 622, 623, 625, 626, 627, 628, 629, 631, 632, 633, 634, 635, 637, 638, 639, 641, 643, 645, 646, 647, 649, 651, 652, 653, 655, 656, 657, 658, 659, 661, 662, 663, 664, 665, 667, 668, 669, 670, 671, 673, 674, 675, 676, 677, 679, 681, 682, 683, 685, 686, 687, 688, 689, 691, 692, 693, 694, 695, 697, 698, 699, 701, 703, 705, 706, 707, 709, 710, 711, 712, 713, 715, 716, 717, 718, 719, 721, 722, 723, 724, 725, 727, 729, 730, 731, 733, 734, 735, 737, 739, 741, 742, 743, 745, 746, 747, 749, 751, 752, 753, 754, 755, 757, 758, 759, 761, 763, 764, 765, 766, 767, 769, 771, 772, 773, 775, 776, 777, 778, 779, 781, 782, 783, 785, 787, 788, 789, 790, 791, 793, 794, 795, 796, 797, 799, 801, 802, 803, 805, 806, 807, 808, 809, 811, 813, 814, 815, 817, 818, 819, 821, 823, 824, 825, 826, 827, 829, 830, 831, 833, 835, 837, 838, 839, 841, 842, 843, 844, 845, 847, 848, 849, 850, 851, 853, 854, 855, 856, 857, 859, 861, 862, 863, 865, 866, 867, 869, 871, 872, 873, 874, 875, 877, 878, 879, 881, 883, 884, 885, 886, 887, 889, 890, 891, 892, 893, 895, 897, 898, 899, 901, 902, 903, 904, 905, 907, 908, 909, 911, 913, 914, 915, 916, 917, 919, 921, 922, 923, 925, 926, 927, 929, 931, 932, 933, 934, 935, 937, 938, 939, 941, 943, 944, 946, 947, 949, 950, 951, 953, 955, 956, 957, 958, 959, 961, 962, 963, 964, 965, 967, 969, 970, 971, 973, 974, 975, 976, 977, 979, 981, 982, 983, 985, 986, 987, 988, 989, 991, 993, 994, 995, 997, 998, 999, 1,001, 1,003, 1,004, 1,005, 1,006, 1,007, 1,009, 1,010, 1,011, 1,012, 1,013, 1,015, 1,016, 1,017, 1,018, 1,019, 1,021, 1,022, 1,023, 1,024, 1,025, 1,027, 1,028, 1,029, 1,030, 1,031, 1,033, 1,034, 1,035, 1,037, 1,039, 1,041, 1,042, 1,043, 1,045, 1,046, 1,047, 1,048, 1,049, 1,051, 1,052, 1,053, 1,054, 1,055, 1,057, 1,058, 1,059, 1,061, 1,063, 1,065, 1,066, 1,067, 1,069, 1,070, 1,071, 1,072, 1,073, 1,075, 1,076, 1,077, 1,078, 1,079, 1,081, 1,082, 1,083, 1,084, 1,085, 1,087, 1,089, 1,090, 1,091, 1,093, 1,094, 1,095, 1,096, 1,097, 1,099, 1,101, 1,102, 1,103, 1,105, 1,106, 1,107, 1,108, 1,109, 1,111, 1,112, 1,113, 1,114, 1,115, 1,117, 1,118, 1,119, 1,121, 1,123, 1,124, 1,125, 1,126, 1,127, 1,129, 1,130, 1,131, 1,132, 1,133, 1,135, 1,136, 1,137, 1,138, 1,139, 1,141, 1,142, 1,143, 1,145, 1,147, 1,149, 1,150, 1,151, 1,153, 1,154, 1,155, 1,156, 1,157, 1,159, 1,161, 1,162, 1,163, 1,165, 1,166, 1,167, 1,168, 1,169, 1,171, 1,172, 1,173, 1,174, 1,175, 1,177, 1,178, 1,179, 1,181, 1,183, 1,185, 1,186, 1,187, 1,189, 1,191, 1,192, 1,193, 1,195, 1,196, 1,197, 1,198, 1,199, 1,201, 1,202, 1,203, 1,205, 1,207, 1,208, 1,209, 1,210, 1,211, 1,213, 1,214, 1,215, 1,217, 1,219, 1,221, 1,222, 1,223, 1,225, 1,226, 1,227, 1,228, 1,229, 1,231, 1,233, 1,234, 1,235, 1,237, 1,238, 1,239, 1,241, 1,243, 1,244, 1,245, 1,246, 1,247, 1,249, 1,250, 1,251, 1,252, 1,253, 1,255, 1,256, 1,257, 1,258, 1,259, 1,261, 1,262, 1,263, 1,264, 1,265, 1,267, 1,268, 1,269, 1,270, 1,271, 1,273, 1,274, 1,275, 1,276, 1,277, 1,279, 1,281, 1,282, 1,283, 1,285, 1,286, 1,287, 1,289, 1,291, 1,292, 1,293, 1,294, 1,295, 1,297, 1,298, 1,299, 1,301, 1,303, 1,304, 1,305, 1,306, 1,307, 1,309, 1,310, 1,311, 1,313, 1,315, 1,317, 1,318, 1,319, 1,321, 1,322, 1,323, 1,324, 1,325, 1,327

以下の解説記事はまだ翻訳されておらず、英語で表示されます。

不足数について

Deficiency is the common case, and for most of its history it was treated as the uninteresting one. Nicomachus of Gerasa introduced the idea around 100 CE in his Introduction to Arithmetic, as one of three classes into which he divided the integers according to how the sum of a number's divisors compares with the number itself. His term for this class translates as "falling short", and it belonged to a moral scheme in which abundance was excess, deficiency was lack, and perfection sat as the virtuous mean between them. Boethius rendered it numerus deficiens in De institutione arithmetica around 500 CE, and because that book was a set text of the medieval quadrivium the vocabulary survived into English largely unaltered.

What makes deficiency structurally interesting is that it is inherited downwards. Every divisor of a perfect or deficient number is itself deficient, which pulls in every prime, every power of a prime, and every factor of a perfect number. Primes are as deficient as a number can be for its size: a prime p has only the divisor 1 beneath it, so it falls short by p − 1.

The class acquired genuine open problems through aliquot sequences. Replacing a number repeatedly by the sum of its proper divisors gives a trajectory that either dies at 1, settles into a perfect or amicable or sociable cycle, or — as far as anyone can prove — might grow without limit. Eugène Catalan raised the question in 1888 and Leonard Eugene Dickson sharpened it in 1913; the Catalan–Dickson conjecture, that every such sequence terminates or becomes periodic, is still open. The smallest unresolved starting value is 276, pushed for decades without a verdict.

Powers of two mark the boundary of the class. The proper divisors of 2^k sum to 2^k − 1, falling short by exactly one, which makes every power of two "almost perfect". Whether any other almost perfect number exists remains unknown.

主な性質

  • A number is deficient when its proper divisors sum to less than itself, equivalently σ(n) < 2n. The first few are 1, 2, 3, 4, 5, 7, 8 and 9.
  • Deficiency is the common case: roughly 75.2% of integers are deficient, the complement of the roughly 24.8% that are abundant, since perfect numbers have density zero.
  • Every prime and every power of a prime is deficient. A prime p falls short by p − 1, the largest shortfall possible at that size.
  • Every divisor of a perfect or deficient number is itself deficient.
  • Every power of two is "almost perfect", falling short by exactly 1: the proper divisors of 256 sum to 255.
  • No almost perfect number other than a power of two has ever been found, and whether one exists is an open problem.
  • Every odd number below 945 is deficient. 945 = 3³ × 5 × 7 is the smallest odd number that is not.
  • 1 is deficient: it has no proper divisors, so its divisor sum is 0.

ほかの個数

出典