Don't take life too seriously: it's just a temporary situation

Smith (numeri di)

Rappresentazione dei numeri  Vari 

Indice

  1. 1. Pagina principale
  2. 2. Insiemi di Monica e di Suzanne

I numeri composti per i quali la differenza tra la somma delle cifre dei fattori primi e la somma delle cifre è un multiplo di n formano l’insieme di Monica Mn; M1 è quindi l’insieme dei numeri composti. Dato che nel caso dei numeri di Smith la differenza è zero, essi sono membri di tutti gli insiemi di Monica.

 

Per ogni n, Mn ha infiniti membri.

 

La tabella seguente mostra gli elementi fino a 1000 degli insiemi di Monica da M2 a M20.

n

Mn

2

4, 8, 10, 12, 14, 15, 22, 26, 27, 35, 42, 44, 45, 54, 56, 58, 60, 62, 63, 64, 65, 68, 78, 84, 85, 88, 90, 92, 94, 96, 99, 102, 108, 111, 118, 119, 121, 122, 123, 126, 129, 133, 136, 138, 141, 143, 145, 152, 155, 158, 159, 160, 161, 164, 165, 166, 169, 174, 175, 176, 177, 180, 182, 183, 184, 186, 187, 188, 189, 194, 196, 198, 200, 202, 206, 207, 208, 210, 215, 216, 231, 232, 235, 238, 240, 242, 243, 245, 246, 248, 250, 252, 255, 261, 265, 273, 274, 275, 278, 279, 280, 282, 284, 286, 288, 294, 300, 301, 302, 305, 308, 316, 319, 320, 323, 325, 328, 329, 330, 334, 336, 338, 340, 341, 344, 346, 351, 352, 355, 360, 361, 364, 368, 371, 374, 377, 378, 380, 382, 384, 385, 386, 390, 391, 392, 395, 396, 400, 402, 404, 405, 406, 411, 414, 415, 417, 422, 423, 424, 428, 429, 432, 434, 438, 442, 447, 452, 453, 454, 455, 456, 458, 460, 462, 466, 468, 471, 474, 475, 476, 477, 480, 483, 484, 485, 489, 490, 500, 504, 506, 510, 511, 512, 513, 514, 515, 516, 517, 520, 522, 524, 525, 526, 527, 528, 529, 533, 534, 535, 548, 550, 551, 552, 553, 554, 558, 559, 562, 565, 566, 570, 572, 574, 575, 576, 578, 581, 582, 583, 588, 589, 592, 596, 602, 603, 605, 608, 609, 616, 618, 627, 634, 636, 637, 638, 639, 644, 645, 646, 648, 651, 654, 655, 656, 657, 660, 663, 666, 668, 674, 682, 684, 690, 692, 695, 700, 702, 703, 706, 708, 713, 714, 715, 717, 720, 723, 726, 728, 729, 731, 732, 734, 737, 740, 742, 744, 746, 750, 753, 756, 760, 762, 764, 767, 768, 770, 771, 776, 778, 779, 781, 782, 783, 785, 788, 789, 792, 793, 798, 799, 800, 801, 804, 805, 806, 808, 810, 816, 818, 820, 822, 824, 825, 833, 844, 845, 847, 848, 852, 858, 860, 861, 862, 866, 867, 870, 873, 874, 875, 876, 882, 884, 889, 891, 892, 894, 895, 896, 897, 898, 901, 902, 904, 908, 909, 910, 913, 914, 915, 922, 923, 925, 926, 927, 928, 930, 932, 935, 936, 938, 943, 946, 948, 949, 950, 954, 956, 958, 961, 963, 964, 965, 966, 975, 979, 980, 981, 985, 986, 989, 992, 996, 998, 999, 1000

3

4, 9, 10, 22, 24, 25, 27, 34, 42, 46, 55, 58, 60, 72, 78, 81, 82, 85, 94, 105, 106, 114, 115, 118, 121, 126, 128, 132, 142, 145, 150, 166, 178, 180, 186, 187, 192, 195, 202, 204, 205, 214, 216, 222, 224, 226, 231, 234, 235, 243, 253, 256, 258, 262, 265, 274, 276, 285, 289, 295, 298, 315, 319, 320, 330, 334, 336, 342, 346, 348, 355, 357, 358, 366, 375, 378, 382, 391, 392, 394, 396, 402, 415, 416, 429, 438, 445, 448, 450, 451, 454, 465, 466, 474, 478, 480, 483, 492, 493, 502, 505, 510, 514, 517, 526, 529, 535, 538, 540, 555, 558, 560, 562, 564, 565, 576, 582, 583, 585, 586, 588, 608, 609, 612, 618, 622, 624, 627, 634, 636, 640, 645, 648, 649, 654, 655, 663, 666, 667, 685, 686, 690, 693, 694, 697, 702, 704, 706, 708, 718, 726, 728, 729, 745, 762, 766, 774, 778, 781, 784, 799, 800, 802, 825, 828, 832, 834, 835, 838, 840, 841, 852, 855, 861, 862, 865, 870, 886, 895, 897, 898, 901, 906, 912, 913, 915, 922, 934, 942, 943, 945, 955, 958, 969, 978, 979, 980, 982, 985, 987, 990, 992, 996

4

4, 12, 14, 22, 27, 35, 58, 63, 85, 88, 90, 94, 108, 119, 121, 122, 136, 141, 152, 158, 159, 160, 161, 166, 169, 174, 175, 176, 194, 198, 202, 215, 238, 245, 248, 252, 255, 261, 265, 274, 279, 280, 282, 286, 294, 302, 305, 319, 320, 325, 329, 338, 346, 351, 355, 360, 368, 371, 378, 380, 382, 391, 395, 402, 405, 411, 414, 417, 423, 424, 432, 438, 442, 454, 456, 460, 468, 471, 475, 477, 483, 484, 485, 489, 490, 510, 517, 520, 525, 526, 535, 550, 554, 559, 562, 570, 572, 576, 581, 588, 589, 602, 627, 634, 636, 639, 645, 646, 648, 654, 656, 663, 666, 682, 690, 706, 714, 728, 729, 731, 734, 740, 742, 750, 753, 760, 762, 778, 779, 792, 793, 800, 804, 805, 806, 808, 816, 820, 822, 824, 825, 833, 844, 845, 847, 848, 852, 861, 866, 873, 876, 882, 884, 889, 891, 892, 895, 897, 910, 913, 914, 915, 922, 925, 927, 946, 949, 950, 958, 961, 963, 964, 966, 975, 985, 999, 1000

5

4, 22, 27, 32, 39, 51, 58, 66, 77, 80, 85, 86, 93, 94, 99, 102, 104, 111, 121, 133, 135, 144, 147, 153, 162, 166, 202, 213, 228, 232, 240, 250, 260, 265, 266, 267, 272, 274, 287, 301, 309, 316, 319, 321, 323, 340, 346, 355, 361, 363, 378, 382, 387, 391, 406, 408, 438, 440, 444, 446, 454, 464, 473, 482, 483, 495, 497, 498, 504, 511, 512, 513, 517, 518, 522, 526, 534, 535, 536, 537, 544, 549, 551, 553, 562, 568, 576, 579, 588, 591, 594, 603, 606, 620, 621, 626, 627, 633, 634, 636, 645, 648, 650, 654, 662, 663, 665, 666, 671, 672, 680, 687, 690, 698, 703, 706, 707, 720, 728, 729, 738, 741, 748, 749, 752, 755, 762, 775, 778, 789, 791, 795, 807, 810, 825, 837, 842, 849, 852, 856, 858, 861, 869, 878, 888, 895, 903, 909, 913, 915, 917, 920, 922, 944, 952, 957, 958, 959, 960, 985, 994, 1000

6

4, 10, 22, 27, 42, 58, 60, 78, 85, 94, 118, 121, 126, 145, 166, 180, 186, 187, 202, 216, 231, 235, 243, 265, 274, 319, 320, 330, 334, 336, 346, 355, 378, 382, 391, 392, 396, 402, 415, 429, 438, 454, 466, 474, 480, 483, 510, 514, 517, 526, 529, 535, 558, 562, 565, 576, 582, 583, 588, 608, 609, 618, 627, 634, 636, 645, 648, 654, 655, 663, 666, 690, 702, 706, 708, 726, 728, 729, 762, 778, 781, 799, 800, 825, 852, 861, 862, 870, 895, 897, 898, 901, 913, 915, 922, 943, 958, 979, 980, 985, 992, 996

7

4, 20, 21, 22, 27, 30, 40, 50, 58, 69, 70, 85, 94, 110, 116, 121, 125, 130, 134, 166, 170, 171, 190, 200, 202, 210, 212, 220, 221, 225, 230, 236, 237, 265, 270, 274, 290, 291, 297, 300, 306, 310, 312, 314, 319, 322, 324, 332, 333, 335, 339, 346, 354, 355, 356, 369, 370, 378, 382, 391, 393, 400, 410, 425, 430, 435, 438, 454, 470, 472, 483, 488, 496, 500, 517, 526, 530, 535, 562, 576, 580, 588, 590, 610, 611, 625, 627, 634, 636, 645, 648, 654, 663, 666, 670, 676, 688, 690, 700, 705, 706, 710, 716, 725, 728, 729, 730, 735, 762, 778, 790, 801, 825, 826, 830, 831, 850, 852, 861, 868, 879, 890, 895, 913, 915, 922, 933, 940, 958, 970, 976, 984, 985, 988, 989

8

4, 22, 27, 58, 85, 88, 94, 121, 141, 152, 160, 166, 169, 198, 202, 245, 252, 261, 265, 274, 279, 280, 286, 319, 346, 355, 360, 378, 380, 382, 391, 405, 411, 423, 432, 438, 454, 483, 484, 489, 490, 517, 520, 525, 526, 535, 559, 562, 570, 576, 588, 589, 602, 627, 634, 636, 645, 648, 654, 663, 666, 682, 690, 706, 714, 728, 729, 740, 750, 760, 762, 778, 779, 793, 804, 808, 825, 833, 844, 847, 852, 861, 866, 889, 892, 895, 910, 913, 915, 922, 927, 946, 949, 950, 958, 961, 964, 985, 999

9

4, 22, 27, 58, 85, 94, 105, 114, 121, 150, 166, 202, 204, 222, 224, 265, 274, 315, 319, 342, 346, 355, 378, 382, 391, 438, 445, 450, 454, 483, 517, 526, 535, 540, 560, 562, 576, 588, 612, 627, 634, 636, 640, 645, 648, 654, 663, 666, 690, 697, 706, 728, 729, 762, 778, 825, 840, 841, 852, 861, 886, 895, 912, 913, 915, 922, 958, 978, 985

10

4, 22, 27, 58, 85, 94, 99, 102, 111, 121, 133, 166, 202, 232, 240, 250, 265, 274, 301, 316, 319, 323, 340, 346, 355, 361, 378, 382, 391, 406, 438, 454, 483, 504, 511, 512, 513, 517, 522, 526, 534, 535, 551, 553, 562, 576, 588, 603, 627, 634, 636, 645, 648, 654, 663, 666, 690, 703, 706, 720, 728, 729, 762, 778, 789, 810, 825, 852, 858, 861, 895, 909, 913, 915, 922, 958, 985, 1000

11

4, 22, 27, 58, 85, 94, 112, 120, 121, 140, 166, 202, 265, 274, 299, 304, 319, 343, 346, 350, 355, 378, 382, 391, 438, 441, 454, 483, 501, 517, 526, 531, 532, 535, 562, 576, 588, 598, 627, 630, 632, 634, 636, 645, 648, 654, 663, 666, 669, 678, 689, 690, 706, 722, 728, 729, 759, 762, 778, 786, 812, 825, 852, 861, 895, 899, 900, 913, 915, 922, 931, 939, 958, 985, 993

12

4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 320, 346, 355, 378, 382, 391, 402, 438, 454, 483, 510, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 800, 825, 852, 861, 895, 897, 913, 915, 922, 958, 985

13

4, 22, 27, 58, 85, 94, 100, 121, 166, 201, 202, 203, 265, 274, 319, 346, 355, 378, 382, 391, 413, 420, 438, 454, 483, 517, 526, 535, 562, 576, 588, 600, 623, 627, 634, 636, 645, 648, 654, 663, 666, 690, 699, 706, 711, 712, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 968, 985

14

4, 22, 27, 58, 85, 94, 121, 166, 200, 202, 210, 265, 274, 300, 319, 346, 355, 378, 382, 391, 400, 438, 454, 483, 500, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 700, 706, 728, 729, 762, 778, 801, 825, 852, 861, 895, 913, 915, 922, 958, 985, 989

15

4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985

16

4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985

17

4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985

18

4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985

19

4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985

20

4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985, 1000

 

L’insieme di Suzanne Sn è invece l’insieme dei numeri composti per i quali sia la somma delle cifre, che la somma delle cifre dei fattori primi sono multiple di n. S1 è quindi l’insieme dei numeri composti.

Dato che nel caso dei numeri di Smith la differenza è zero, i numeri di Smith che sono anche harshad sono membri di tutti gli insiemi di Suzanne, ognuno dei quali contiene quindi infiniti elementi.

 

Per ogni n, Sn è un sottoinsieme di Mn.

 

La tabella seguente mostra gli elementi fino a 1000 degli insiemi di Suzanne da S2 a S20.

n

Sn

2

4, 8, 15, 22, 26, 35, 42, 44, 60, 62, 64, 68, 84, 88, 99, 118, 121, 123, 129, 136, 138, 141, 143, 145, 152, 158, 161, 165, 169, 174, 176, 183, 187, 189, 194, 196, 198, 200, 202, 206, 208, 215, 231, 235, 240, 242, 246, 248, 255, 273, 275, 279, 280, 282, 284, 286, 288, 301, 305, 316, 323, 325, 329, 330, 334, 336, 338, 341, 352, 361, 374, 378, 385, 390, 392, 396, 400, 402, 404, 406, 411, 415, 417, 422, 424, 428, 442, 453, 455, 460, 462, 466, 468, 471, 475, 477, 480, 484, 510, 512, 514, 516, 525, 527, 529, 534, 550, 552, 554, 558, 565, 570, 572, 574, 576, 578, 581, 583, 589, 592, 596, 602, 608, 637, 639, 644, 646, 648, 651, 655, 657, 660, 666, 668, 682, 684, 695, 703, 714, 723, 729, 732, 734, 750, 756, 767, 770, 776, 778, 781, 783, 785, 789, 792, 798, 800, 804, 806, 808, 820, 822, 824, 833, 844, 848, 860, 862, 866, 873, 875, 882, 884, 891, 895, 897, 901, 909, 910, 914, 923, 925, 927, 930, 932, 936, 938, 943, 949, 950, 954, 956, 958, 961, 963, 965, 981, 985, 989, 992, 996, 998

3

9, 24, 27, 42, 60, 72, 78, 81, 105, 114, 126, 132, 150, 180, 186, 192, 195, 204, 216, 222, 231, 234, 243, 258, 276, 285, 315, 330, 336, 342, 348, 357, 366, 375, 378, 396, 402, 429, 438, 450, 465, 474, 480, 483, 492, 510, 540, 555, 558, 564, 576, 582, 585, 588, 609, 612, 618, 624, 627, 636, 645, 648, 654, 663, 666, 690, 693, 702, 708, 726, 729, 762, 774, 825, 828, 834, 840, 852, 855, 861, 870, 897, 906, 912, 915, 942, 945, 969, 978, 987, 990, 996

4

4, 22, 35, 88, 121, 152, 161, 169, 174, 202, 215, 255, 282, 286, 305, 417, 471, 475, 484, 525, 570, 602, 646, 682, 714, 750, 800, 804, 808, 822, 844, 848, 866, 884, 897, 925, 961

5

32, 104, 267, 316, 361, 406, 438, 483, 497, 537, 591, 627, 636, 645, 654, 663, 672, 690, 703, 749, 762, 807, 825, 852, 861, 915, 960

6

42, 60, 231, 330, 336, 378, 396, 402, 480, 510, 558, 576, 648, 666, 729, 897, 996

7

70, 322, 356, 430, 588, 590, 610, 700, 716, 725, 984

8

88, 152, 169, 286, 484, 602, 682, 808, 844, 961

9

27, 315, 342, 378, 450, 540, 576, 612, 648, 666, 729

10

316, 361, 406, 703

11

598, 632, 722, 778, 812, 895, 958, 985

12

897

13

58, 85, 94, 166, 265, 274, 319, 346, 355, 382, 391, 454, 517, 526, 535, 562, 634, 706, 913, 922

14

-

15

438, 483, 627, 636, 645, 654, 663, 690, 762, 825, 852, 861, 915

16

-

17

728

18

378, 576, 648, 666, 7292

19

-

20

-

 

Bibliografia

  • Eves, Howard W.;  Mathematical Circles, Mathematical Association of America, vol. III, 2003 -

    Una stupenda raccolta di aneddoti a sfondo matematico, pubblicati inizialmente con i titoli Mathematical Circles Adieu (Prindle, Weber and Schmidt Inc., 1977) e Return to Mathematical Circles (Prindle, Weber and Schmidt Inc., 1988).

  • Guy, Richard K.;  Woodrow, Robert E.;  The Lighter Side of Mathematics: Proceedings of the Eugene Strens Memorial Conference on Recreational Mathematics and Its History, Washington, Mathematical Association of America, 1994.
  • Pickover, Clifford A.;  The Wonders of Numbers, New York, Oxford University Press, 2001.
  • Roberts, Joe;  The Lure of the Integers, The Mathematical Association of America, 1992 -

    Una miniera di informazioni sugli interi.

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