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

Piramidali centrati (numeri)

Numeri figurati 

Indice

  1. 1. Pagina principale
  2. 2. Espressione di interi come somma di numeri piramidali centrati

Secondo una mia congettura ogni intero positivo si può esprimere come somma di n + 3 numeri piramidali centrati a n lati, tranne nel caso n = 3, nel quale servono probabilmente 7 addendi. Per interi abbastanza grandi ne bastano sicuramente 8, e probabilmente 4.

 

La tabella seguente riporta il numero di interi non rappresentabili come somma di 4 numeri piramidali centrati e il massimo noto, per piramidi fino a 10 lati; se ve ne sono altri, sono maggiori di 1010 per le piramidi fino a 7 lati e di 3 • 1010 per quelle con 8 o più lati (M. Fiorentini, 2016).

Lati

Numero di interi non rappresentabili come somma di 4 numeri piramidali centrati

Massimo intero non rappresentabile come somma di 4 numeri piramidali centrati

3

30313

74968967

4

46431

v. numeri ottaedrici

5

112557

183479498

6

v. cubi

v. cubi

7

486532

999033049

8

2350202

17714658187

9

1383380

4582269623

10

2193998

5550232306

Per gli interi che richiedono più di 4 numeri piramidali centrati a 4 lati v. numeri ottaedrici.

Per gli interi che richiedono più di 4 numeri piramidali centrati a 6 lati v. cubi.

 

Qui trovate gli interi minori di 1010 che richiedono più di 4 numeri piramidali centrati a 3 lati, ciascuno seguito dal numero di addendi necessari.

Qui trovate gli interi minori di 1010 che richiedono più di 4 numeri piramidali centrati a 4 lati, ciascuno seguito dal numero di addendi necessari.

Qui trovate gli interi minori di 1010 che richiedono più di 4 numeri piramidali centrati a 5 lati, ciascuno seguito dal numero di addendi necessari (1.1 Mbyte).

Qui trovate gli interi minori di 1010 che richiedono più di 4 numeri piramidali centrati a 7 lati, ciascuno seguito dal numero di addendi necessari (5 Mbyte).

Qui trovate gli interi minori di 106 che richiedono più di 4 numeri piramidali centrati a 8 lati, ciascuno seguito dal numero di addendi necessari (2.4 Mbyte).

Qui trovate gli interi minori di 106 che richiedono più di 4 numeri piramidali centrati a 9 lati, ciascuno seguito dal numero di addendi necessari (2.5 Mbyte).

Qui trovate gli interi minori di 106 che richiedono più di 4 numeri piramidali centrati a 10 lati, ciascuno seguito dal numero di addendi necessari (3 Mbyte).

 

La tabella seguente riporta il numero di interi che richiedono 5 addendi e il massimo noto, per piramidi fino a 10 lati; se ve ne sono altri, sono maggiori di 1010 per le piramidi fino a 7 lati e di 3 • 1010 per quelle con 8 o più lati (M. Fiorentini, 2016).

Lati

Numero di interi che richiedono 5 addendi

Massimo intero che richiede 5 addendi

3

30239

74968967

4

46251

v. numeri ottaedrici

5

111818

183479498

6

v. cubi

v. cubi

7

484117

999033049

8

2347132

17714658187

9

1379895

4582269623

10

2189165

5550232306

 

La tabella seguente riporta il numero di interi che richiedono 6 addendi e il massimo noto, per piramidi fino a 10 lati; se ve ne sono altri, sono maggiori di 1010 per le piramidi fino a 7 lati e di 3 • 1010 per quelle con 8 o più lati (M. Fiorentini, 2016).

Lati

Numero di interi che richiedono 6 addendi

Massimo intero che richiede 6 addendi

3

71

8107

4

168

11579

5

685

29124

6

3922

1290740

7

2194

119415

8

2813

313847

9

3158

174492

10

4380

375182

 

La tabella seguente riporta gli interi fino a 1000 che richiedono 5 addendi, per piramidi centrate fino a 10 lati.

Lati

Interi che richiedono 5 addendi

3

9, 13, 19, 23, 27, 33, 38, 42, 47, 52, 56, 61, 77, 87, 89, 91, 93, 94, 97, 106, 108, 120, 123, 124, 125, 137, 139, 143, 144, 149, 152, 153, 157, 162, 163, 166, 168, 170, 171, 172, 174, 183, 187, 189, 193, 197, 199, 201, 203, 207, 212, 216, 218, 221, 226, 230, 233, 234, 235, 247, 249, 251, 253, 254, 259, 264, 268, 272, 273, 278, 282, 284, 293, 297, 303, 307, 311, 313, 315, 317, 319, 322, 323, 332, 336, 342, 344, 346, 349, 353, 357, 358, 361, 363, 368, 378, 382, 388, 392, 396, 407, 411, 417, 421, 425, 427, 428, 429, 432, 443, 446, 447, 448, 453, 456, 457, 458, 459, 460, 463, 467, 472, 475, 477, 478, 479, 488, 489, 491, 492, 498, 509, 513, 517, 518, 523, 524, 527, 531, 537, 538, 542, 552, 553, 556, 562, 563, 566, 567, 568, 577, 581, 582, 584, 587, 589, 594, 595, 597, 598, 599, 601, 603, 608, 613, 620, 623, 627, 633, 637, 641, 647, 648, 652, 653, 654, 658, 661, 662, 666, 679, 683, 693, 698, 703, 704, 708, 712, 717, 718, 722, 723, 726, 730, 731, 733, 747, 749, 757, 762, 763, 764, 769, 776, 778, 786, 789, 790, 793, 794, 807, 811, 813, 815, 820, 822, 823, 824, 826, 827, 829, 832, 834, 836, 839, 840, 842, 843, 844, 853, 857, 859, 863, 865, 867, 878, 882, 888, 892, 896, 899, 903, 907, 917, 921, 925, 929, 930, 948, 949, 952, 956, 959, 963, 967, 968, 971, 975, 977, 988, 989, 992, 993, 994, 995

4

5, 10, 15, 23, 28, 30, 33, 41, 43, 48, 53, 59, 66, 68, 71, 77, 81, 84, 96, 99, 102, 109, 112, 114, 117, 119, 120, 122, 125, 127, 134, 137, 139, 143, 144, 145, 150, 155, 157, 160, 161, 168, 175, 178, 180, 183, 186, 188, 193, 198, 201, 204, 211, 216, 218, 221, 222, 223, 226, 227, 229, 236, 241, 245, 246, 247, 254, 259, 263, 264, 267, 268, 271, 272, 279, 280, 283, 284, 289, 296, 297, 301, 302, 305, 306, 307, 310, 314, 321, 324, 326, 329, 331, 332, 334, 339, 343, 348, 349, 353, 358, 359, 367, 368, 371, 372, 373, 374, 376, 381, 385, 386, 391, 398, 399, 403, 405, 406, 409, 410, 414, 416, 419, 423, 424, 425, 428, 434, 437, 442, 446, 447, 450, 452, 455, 458, 459, 460, 466, 470, 471, 475, 477, 480, 483, 484, 485, 488, 493, 498, 503, 505, 511, 513, 516, 518, 519, 521, 526, 530, 531, 536, 541, 543, 544, 549, 551, 554, 556, 559, 561, 564, 565, 568, 569, 572, 573, 579, 585, 588, 589, 590, 592, 597, 601, 602, 603, 605, 606, 610, 611, 615, 622, 626, 628, 630, 631, 633, 639, 640, 643, 644, 646, 648, 651, 653, 656, 657, 658, 664, 667, 668, 674, 675, 681, 684, 686, 692, 697, 702, 705, 706, 710, 717, 718, 719, 724, 728, 729, 730, 731, 735, 741, 742, 743, 746, 747, 748, 750, 753, 760, 763, 766, 768, 771, 784, 785, 786, 788, 789, 794, 795, 796, 797, 798, 801, 803, 809, 813, 814, 815, 819, 820, 821, 826, 827, 829, 830, 831, 837, 842, 844, 847, 851, 855, 856, 862, 863, 864, 865, 868, 869, 873, 880, 886, 887, 889, 895, 900, 905, 914, 915, 923, 928, 931, 932, 933, 934, 940, 943, 944, 949, 953, 956, 958, 961, 965, 966, 967, 969, 971, 974, 975, 991, 993, 994, 999, 1000

5

5, 11, 17, 27, 29, 33, 35, 39, 45, 49, 51, 58, 64, 67, 71, 80, 82, 83, 86, 93, 98, 99, 102, 111, 117, 121, 124, 127, 133, 137, 139, 143, 145, 146, 149, 153, 155, 164, 165, 168, 170, 175, 176, 177, 180, 186, 191, 192, 197, 203, 207, 208, 209, 215, 219, 221, 223, 225, 229, 231, 238, 239, 241, 244, 247, 251, 257, 260, 263, 266, 268, 270, 272, 273, 274, 278, 279, 282, 291, 297, 303, 304, 307, 313, 319, 321, 323, 325, 329, 331, 335, 339, 344, 345, 350, 354, 357, 361, 366, 372, 377, 378, 379, 383, 388, 397, 401, 403, 405, 407, 409, 410, 413, 419, 421, 424, 425, 427, 432, 440, 441, 444, 448, 450, 454, 456, 460, 461, 462, 466, 471, 472, 477, 485, 488, 493, 499, 501, 503, 507, 511, 513, 515, 519, 524, 525, 527, 530, 538, 542, 546, 548, 552, 554, 558, 560, 564, 567, 568, 570, 577, 583, 586, 589, 591, 595, 599, 601, 603, 605, 613, 619, 621, 625, 626, 631, 636, 637, 642, 643, 644, 647, 652, 653, 657, 658, 666, 668, 669, 673, 674, 681, 683, 684, 685, 688, 689, 694, 695, 699, 700, 701, 704, 705, 707, 710, 719, 725, 726, 730, 731, 732, 734, 735, 736, 741, 746, 747, 750, 751, 752, 753, 757, 763, 766, 767, 772, 777, 779, 782, 783, 785, 787, 793, 794, 799, 801, 805, 807, 808, 810, 811, 815, 816, 824, 825, 828, 829, 831, 832, 833, 834, 839, 846, 848, 852, 853, 869, 871, 875, 876, 877, 883, 885, 887, 892, 893, 899, 905, 907, 909, 914, 916, 921, 922, 927, 928, 931, 932, 934, 937, 938, 945, 952, 955, 956, 958, 959, 965, 967, 969, 974, 975, 977, 979, 980, 985, 987, 989, 990, 991, 993, 996, 998

6

5, 12, 19, 26, 31, 33, 38, 40, 45, 52, 57, 59, 68, 71, 75, 78, 83, 90, 94, 96, 97, 101, 109, 115, 116, 120, 131, 138, 143, 146, 150, 157, 162, 164, 169, 171, 172, 176, 181, 183, 188, 194, 195, 199, 201, 207, 208, 209, 213, 214, 220, 227, 234, 241, 246, 248, 255, 257, 260, 262, 264, 267, 269, 272, 274, 279, 283, 286, 290, 293, 298, 305, 309, 312, 316, 318, 320, 323, 324, 325, 330, 331, 335, 347, 354, 358, 361, 365, 373, 377, 380, 381, 384, 387, 391, 394, 396, 399, 403, 410, 414, 417, 421, 422, 425, 429, 431, 436, 443, 447, 449, 450, 451, 456, 462, 473, 475, 478, 480, 482, 485, 487, 488, 492, 494, 499, 501, 505, 506, 508, 511, 516, 524, 525, 527, 531, 534, 537, 538, 542, 543, 544, 549, 550, 556, 562, 563, 564, 569, 573, 579, 582, 583, 588, 595, 597, 599, 600, 602, 605, 609, 612, 614, 618, 619, 622, 625, 628, 629, 632, 642, 647, 651, 654, 655, 658, 660, 661, 666, 668, 673, 676, 677, 680, 683, 689, 690, 693, 696, 699, 700, 703, 705, 710, 715, 716, 717, 719, 720, 722, 726, 733, 741, 747, 749, 752, 754, 759, 761, 766, 767, 768, 771, 774, 778, 780, 781, 785, 790, 796, 797, 799, 803, 804, 807, 808, 813, 815, 816, 817, 818, 822, 827, 829, 834, 836, 837, 840, 841, 843, 846, 848, 859, 866, 869, 872, 873, 874, 876, 878, 879, 885, 888, 891, 892, 895, 897, 898, 901, 904, 907, 911, 914, 916, 922, 925, 928, 930, 934, 935, 937, 939, 948, 951, 955, 956, 960, 962, 963, 964, 967, 969, 974, 984, 985, 986, 989, 990, 992, 993, 995, 996, 998

7

5, 13, 21, 29, 35, 37, 43, 45, 51, 59, 65, 67, 73, 78, 81, 86, 89, 95, 103, 108, 110, 111, 116, 125, 132, 133, 138, 151, 159, 165, 168, 173, 175, 181, 187, 189, 195, 197, 198, 203, 209, 211, 217, 224, 225, 230, 232, 239, 240, 241, 246, 247, 255, 263, 268, 271, 279, 285, 287, 295, 297, 301, 303, 305, 309, 311, 312, 315, 317, 323, 328, 331, 333, 336, 339, 345, 353, 355, 358, 361, 366, 368, 370, 374, 375, 376, 382, 383, 388, 403, 404, 407, 411, 415, 419, 423, 429, 433, 437, 441, 445, 447, 449, 453, 457, 459, 463, 467, 469, 472, 476, 480, 484, 488, 489, 493, 497, 500, 506, 510, 514, 518, 521, 522, 523, 529, 532, 536, 540, 543, 549, 551, 555, 557, 559, 563, 565, 566, 571, 573, 579, 581, 583, 586, 587, 589, 593, 594, 595, 600, 603, 608, 609, 611, 616, 620, 622, 624, 630, 632, 638, 646, 653, 654, 657, 661, 665, 669, 673, 676, 677, 683, 687, 691, 693, 695, 697, 699, 703, 707, 709, 711, 713, 717, 719, 722, 723, 726, 729, 730, 734, 746, 752, 756, 760, 764, 766, 768, 771, 774, 776, 782, 785, 786, 790, 793, 794, 797, 801, 805, 806, 809, 812, 813, 817, 819, 823, 825, 828, 831, 833, 835, 836, 837, 839, 843, 853, 855, 861, 866, 870, 874, 877, 883, 885, 888, 891, 892, 896, 899, 900, 904, 907, 908, 913, 918, 919, 926, 929, 931, 934, 937, 939, 945, 947, 948, 949, 950, 951, 953, 956, 961, 964, 969, 971, 972, 973, 976, 977, 979, 983, 984, 986, 991, 999

8

5, 14, 23, 32, 39, 41, 48, 50, 57, 66, 73, 75, 82, 88, 91, 97, 100, 107, 116, 122, 124, 125, 131, 141, 149, 150, 156, 171, 174, 180, 187, 190, 196, 198, 199, 205, 212, 214, 221, 223, 224, 230, 237, 239, 246, 248, 254, 255, 261, 263, 271, 272, 273, 279, 280, 290, 295, 299, 304, 305, 308, 317, 320, 324, 326, 329, 335, 337, 342, 344, 346, 351, 353, 354, 358, 360, 367, 369, 373, 376, 378, 382, 385, 392, 401, 403, 407, 410, 416, 418, 420, 425, 426, 427, 434, 435, 441, 450, 459, 463, 468, 472, 474, 477, 481, 484, 488, 493, 497, 499, 502, 506, 508, 511, 515, 520, 522, 524, 527, 531, 533, 537, 542, 546, 548, 551, 555, 556, 561, 565, 569, 571, 576, 580, 585, 589, 593, 594, 595, 602, 605, 610, 614, 618, 625, 627, 629, 632, 634, 636, 641, 643, 644, 650, 652, 654, 659, 661, 663, 667, 668, 670, 675, 676, 677, 684, 686, 693, 695, 702, 703, 706, 708, 711, 718, 720, 724, 727, 736, 744, 745, 748, 749, 753, 757, 762, 767, 770, 771, 773, 778, 782, 787, 789, 791, 794, 796, 798, 801, 805, 807, 810, 812, 816, 819, 822, 823, 824, 827, 830, 831, 836, 844, 850, 857, 861, 866, 870, 872, 875, 878, 879, 882, 884, 891, 894, 895, 900, 903, 904, 905, 908, 913, 917, 918, 919, 922, 925, 926, 928, 931, 933, 937, 940, 943, 944, 947, 949, 951, 952, 953, 956, 960, 965, 973, 974, 978, 982, 985, 987, 991, 993, 996, 1000

9

5, 15, 25, 35, 43, 45, 53, 55, 63, 73, 81, 83, 91, 98, 101, 108, 111, 119, 129, 136, 138, 139, 146, 157, 166, 167, 174, 184, 191, 194, 195, 201, 209, 212, 219, 221, 222, 229, 237, 239, 247, 249, 250, 257, 265, 267, 275, 277, 284, 285, 292, 294, 303, 304, 305, 312, 313, 320, 325, 330, 335, 340, 341, 345, 355, 358, 363, 365, 368, 375, 377, 383, 385, 387, 393, 395, 396, 401, 403, 411, 413, 418, 421, 423, 428, 431, 439, 449, 451, 456, 459, 466, 468, 470, 476, 477, 478, 486, 487, 494, 504, 515, 519, 521, 525, 529, 531, 532, 535, 539, 542, 547, 549, 553, 557, 559, 563, 567, 569, 573, 577, 583, 585, 587, 591, 595, 597, 602, 604, 608, 612, 614, 618, 622, 623, 629, 633, 638, 640, 646, 650, 652, 656, 660, 665, 666, 667, 675, 678, 684, 688, 693, 695, 701, 703, 705, 709, 711, 713, 719, 721, 722, 729, 731, 733, 739, 741, 743, 748, 749, 751, 757, 758, 759, 768, 769, 778, 779, 788, 792, 794, 796, 798, 802, 806, 808, 812, 813, 816, 824, 826, 831, 835, 836, 839, 840, 841, 845, 849, 855, 861, 864, 865, 867, 868, 873, 877, 879, 883, 885, 887, 891, 893, 895, 899, 903, 905, 909, 911, 915, 919, 922, 923, 924, 925, 928, 931, 932, 938, 947, 948, 954, 959, 962, 966, 970, 972, 976, 978, 982, 985, 986, 990, 992, 1000

10

5, 16, 27, 38, 47, 49, 58, 60, 69, 80, 89, 91, 100, 108, 111, 119, 122, 131, 142, 150, 152, 153, 161, 173, 183, 184, 192, 203, 211, 214, 215, 222, 231, 234, 242, 244, 245, 253, 262, 264, 273, 275, 276, 284, 293, 295, 304, 306, 314, 315, 323, 325, 335, 336, 337, 345, 346, 354, 360, 365, 367, 371, 376, 377, 382, 393, 396, 398, 402, 404, 407, 415, 417, 424, 426, 428, 435, 437, 438, 444, 446, 455, 457, 459, 463, 466, 468, 474, 477, 486, 497, 499, 505, 508, 516, 518, 520, 527, 528, 529, 538, 539, 547, 558, 560, 566, 571, 575, 577, 582, 586, 588, 589, 593, 597, 600, 606, 608, 613, 617, 619, 621, 624, 628, 630, 635, 639, 646, 648, 650, 655, 659, 661, 667, 669, 670, 674, 678, 680, 685, 689, 690, 697, 701, 707, 709, 711, 716, 720, 722, 727, 731, 737, 738, 739, 748, 751, 758, 762, 768, 770, 777, 779, 781, 786, 788, 790, 797, 799, 800, 808, 810, 812, 819, 821, 823, 829, 830, 832, 839, 840, 841, 852, 863, 871, 874, 878, 880, 882, 885, 889, 894, 896, 900, 901, 902, 905, 913, 916, 921, 922, 926, 927, 930, 931, 932, 937, 941, 948, 955, 958, 959, 961, 962, 963, 968, 972, 974, 979, 981, 983, 988, 990, 992, 997

 

La tabella seguente riporta gli interi fino a 1000 che richiedono 6 addendi, per piramidi centrate fino a 10 lati.

Lati

Interi che richiedono 6 addendi

3

14, 24, 28, 43, 48, 53, 57, 62, 92, 154, 158, 159, 173, 188, 202, 204, 208, 217, 231, 236, 269, 283, 298, 312, 318, 337, 347, 383, 422, 426, 493, 494, 528, 532, 543, 557, 558, 602, 628, 642, 667, 713, 828, 841, 858, 897, 922, 964

4

11, 16, 29, 34, 36, 42, 49, 54, 60, 67, 72, 74, 78, 115, 118, 121, 128, 140, 156, 162, 163, 169, 181, 187, 194, 199, 205, 207, 212, 219, 224, 230, 242, 248, 260, 265, 266, 273, 285, 286, 290, 291, 303, 308, 315, 327, 333, 387, 392, 393, 411, 412, 417, 418, 443, 453, 456, 461, 472, 478, 494, 499, 504, 522, 532, 537, 538, 555, 557, 560, 562, 563, 570, 598, 604, 607, 616, 617, 623, 629, 634, 645, 649, 650, 687, 711, 725, 736, 749, 769, 772, 790, 791, 804, 810, 832, 838, 848, 857, 870, 874, 875, 876, 881, 882, 888, 906, 950, 959, 972

5

6, 12, 18, 34, 36, 40, 42, 50, 52, 59, 65, 72, 74, 81, 87, 89, 90, 94, 103, 118, 125, 134, 140, 144, 147, 150, 156, 171, 172, 178, 187, 193, 198, 199, 200, 222, 226, 230, 232, 245, 246, 248, 252, 254, 261, 262, 269, 275, 277, 280, 283, 285, 292, 298, 305, 314, 320, 326, 327, 328, 330, 336, 346, 351, 352, 358, 367, 368, 373, 375, 380, 384, 389, 390, 404, 411, 412, 426, 433, 434, 455, 457, 463, 464, 473, 478, 479, 486, 494, 495, 502, 508, 510, 516, 518, 526, 531, 532, 539, 549, 553, 555, 561, 565, 569, 571, 578, 584, 592, 593, 600, 606, 608, 614, 622, 654, 659, 660, 667, 675, 676, 690, 691, 696, 697, 698, 706, 708, 711, 712, 720, 727, 742, 748, 749, 758, 759, 764, 773, 774, 780, 786, 788, 789, 795, 800, 802, 806, 812, 817, 818, 840, 841, 847, 854, 855, 878, 894, 900, 906, 908, 923, 929, 930, 939, 946, 953, 960, 976, 981, 982, 988, 992, 999

6

6, 13, 20, 34, 39, 41, 46, 48, 53, 58, 60, 69, 76, 79, 84, 86, 95, 98, 102, 104, 105, 110, 117, 121, 123, 124, 132, 139, 147, 151, 158, 165, 170, 173, 177, 184, 196, 202, 203, 210, 215, 221, 222, 228, 235, 236, 242, 247, 249, 263, 265, 268, 273, 275, 284, 287, 291, 294, 299, 301, 306, 310, 313, 319, 321, 326, 328, 332, 333, 336, 338, 339, 348, 355, 362, 366, 374, 382, 385, 388, 389, 392, 400, 404, 411, 418, 426, 430, 437, 444, 445, 452, 455, 457, 458, 463, 464, 481, 483, 489, 490, 502, 507, 509, 517, 519, 526, 545, 546, 551, 552, 554, 570, 571, 572, 580, 581, 589, 590, 598, 606, 607, 608, 610, 615, 617, 626, 627, 633, 636, 643, 652, 659, 662, 663, 669, 674, 678, 681, 697, 698, 706, 707, 708, 723, 724, 725, 727, 734, 742, 743, 760, 769, 779, 786, 788, 798, 805, 823, 824, 825, 830, 831, 832, 835, 842, 844, 845, 849, 851, 860, 867, 868, 877, 886, 893, 896, 899, 905, 906, 912, 915, 923, 924, 931, 932, 933, 938, 940, 941, 942, 943, 949, 957, 959, 965, 968, 970, 975, 977, 994, 997

7

6, 14, 22, 30, 38, 44, 46, 52, 54, 60, 66, 68, 79, 82, 87, 90, 96, 98, 104, 109, 112, 117, 119, 120, 126, 134, 139, 141, 142, 152, 160, 169, 174, 182, 184, 190, 196, 199, 204, 206, 212, 218, 226, 233, 234, 242, 248, 249, 256, 264, 272, 277, 280, 286, 288, 298, 304, 306, 310, 316, 318, 320, 329, 332, 337, 340, 342, 346, 348, 354, 359, 362, 369, 371, 377, 379, 384, 385, 386, 389, 391, 392, 412, 413, 416, 420, 424, 434, 442, 446, 450, 454, 456, 460, 464, 468, 477, 478, 481, 485, 490, 494, 498, 507, 515, 519, 524, 527, 528, 530, 531, 537, 538, 552, 558, 560, 567, 568, 574, 582, 588, 590, 592, 601, 602, 604, 610, 612, 617, 625, 626, 631, 633, 634, 639, 640, 641, 642, 655, 662, 663, 666, 674, 684, 685, 694, 696, 700, 704, 706, 708, 714, 716, 718, 727, 728, 731, 735, 738, 739, 740, 747, 748, 757, 761, 765, 767, 769, 777, 780, 783, 787, 791, 802, 810, 814, 820, 821, 822, 832, 840, 842, 844, 845, 854, 862, 864, 871, 875, 884, 893, 894, 897, 905, 914, 916, 927, 928, 935, 938, 957, 958, 959, 962, 965, 966, 968, 970, 978, 980, 981, 982, 987, 988, 1000

8

6, 15, 24, 33, 42, 49, 51, 58, 60, 67, 74, 76, 83, 89, 92, 98, 101, 108, 110, 117, 123, 126, 132, 134, 135, 142, 151, 157, 159, 160, 172, 181, 184, 191, 197, 206, 208, 209, 215, 222, 225, 231, 233, 234, 240, 247, 256, 258, 264, 265, 274, 281, 282, 283, 291, 300, 309, 314, 315, 318, 325, 327, 338, 339, 345, 347, 352, 355, 359, 361, 363, 364, 374, 377, 379, 383, 386, 388, 389, 393, 395, 402, 404, 408, 411, 413, 419, 421, 428, 430, 436, 437, 438, 442, 444, 445, 460, 469, 470, 473, 478, 482, 494, 503, 504, 507, 509, 512, 516, 518, 519, 523, 528, 532, 534, 543, 547, 552, 557, 558, 562, 566, 568, 577, 581, 586, 590, 591, 596, 599, 600, 603, 604, 606, 611, 612, 615, 628, 635, 637, 639, 645, 646, 649, 653, 662, 664, 669, 671, 673, 678, 679, 685, 687, 689, 694, 696, 698, 709, 712, 713, 719, 721, 728, 730, 734, 746, 754, 755, 758, 759, 763, 768, 779, 780, 783, 790, 792, 797, 802, 804, 806, 808, 813, 815, 817, 828, 829, 832, 833, 837, 840, 841, 842, 851, 854, 862, 867, 871, 873, 876, 885, 888, 889, 892, 896, 901, 914, 923, 927, 934, 935, 936, 938, 948, 954, 957, 959, 961, 962, 963, 975, 983, 984, 988, 992, 997

9

6, 16, 26, 36, 46, 54, 56, 64, 66, 74, 82, 84, 92, 99, 102, 109, 112, 120, 122, 130, 137, 140, 147, 149, 150, 158, 168, 175, 177, 178, 192, 202, 205, 206, 213, 220, 223, 230, 232, 233, 234, 240, 248, 251, 258, 260, 261, 268, 276, 278, 286, 288, 289, 295, 296, 306, 314, 315, 316, 326, 331, 336, 342, 346, 351, 352, 356, 359, 364, 366, 369, 378, 379, 380, 386, 388, 394, 397, 398, 402, 404, 406, 407, 414, 419, 422, 424, 429, 432, 434, 435, 440, 442, 450, 452, 457, 460, 462, 469, 471, 479, 481, 488, 489, 490, 495, 497, 498, 505, 516, 526, 530, 536, 540, 543, 554, 560, 564, 568, 570, 571, 574, 578, 580, 581, 586, 588, 592, 596, 598, 609, 613, 615, 619, 624, 625, 626, 630, 634, 636, 641, 647, 651, 657, 661, 662, 663, 668, 671, 672, 676, 677, 679, 685, 686, 689, 704, 706, 712, 714, 716, 717, 723, 724, 727, 732, 734, 742, 744, 750, 752, 754, 760, 761, 762, 770, 772, 780, 782, 789, 795, 799, 807, 809, 817, 818, 819, 823, 837, 846, 847, 850, 851, 852, 856, 862, 863, 874, 875, 878, 886, 888, 890, 894, 896, 900, 902, 904, 906, 907, 912, 914, 916, 918, 929, 930, 933, 934, 935, 939, 942, 943, 944, 955, 958, 967, 973, 977, 979, 981, 983, 987, 993, 996, 997, 998

10

6, 17, 28, 39, 50, 59, 61, 70, 72, 81, 90, 92, 101, 103, 109, 112, 120, 123, 132, 134, 143, 151, 154, 162, 164, 165, 174, 185, 193, 195, 196, 204, 212, 216, 223, 226, 227, 235, 243, 246, 254, 256, 257, 258, 265, 274, 277, 285, 287, 288, 296, 305, 307, 316, 318, 319, 326, 327, 338, 347, 348, 349, 361, 366, 372, 378, 379, 383, 388, 389, 394, 397, 403, 405, 408, 418, 419, 420, 427, 429, 436, 439, 440, 441, 445, 447, 449, 450, 458, 464, 467, 469, 471, 475, 478, 480, 481, 487, 489, 498, 500, 502, 506, 509, 511, 519, 521, 530, 532, 540, 541, 542, 548, 550, 551, 559, 572, 578, 583, 587, 590, 594, 598, 601, 609, 614, 620, 625, 629, 631, 632, 633, 636, 640, 642, 643, 649, 651, 656, 660, 662, 664, 675, 679, 681, 682, 686, 691, 692, 693, 698, 702, 704, 710, 717, 721, 723, 728, 732, 733, 734, 740, 743, 744, 749, 750, 752, 754, 759, 760, 763, 765, 771, 780, 782, 789, 791, 793, 794, 801, 802, 805, 811, 813, 822, 824, 826, 831, 833, 835, 842, 843, 844, 853, 855, 864, 866, 875, 881, 883, 886, 890, 895, 897, 906, 908, 912, 914, 928, 933, 938, 939, 942, 943, 944, 945, 949, 956, 969, 970, 973, 975, 982, 984, 986, 991, 993, 998, 1000

 

La tabella seguente riporta gli interi che richiedono 7 addendi, per piramidi centrate fino a 10 lati; se ve ne sono altri, sono maggiori di 1010 per le piramidi fino a 7 lati e di 3 • 1010 per quelle con 8 o più lati (M. Fiorentini, 2016).

Lati

Interi che richiedono 7 addendi

3

29, 58, 63

4

17, 35, 55, 61, 73, 79, 80, 200, 206, 213, 225, 309

5

13, 19, 41, 43, 66, 73, 88, 95, 96, 97, 104, 141, 148, 157, 179, 194, 201, 253, 255, 276, 284, 299, 306, 337, 353, 359, 374, 381, 382, 480, 487, 509, 517, 562, 585, 607, 615, 713, 765, 781, 796, 803, 901, 983, 1035, 1087, 1104, 1268, 1621, 1823, 2132, 4091

6

7, 14, 21, 42, 47, 49, 61, 77, 85, 87, 103, 106, 111, 112, 113, 122, 140, 148, 159, 166, 174, 178, 185, 204, 211, 223, 229, 230, 237, 276, 292, 295, 300, 302, 311, 327, 329, 337, 340, 356, 363, 390, 393, 401, 412, 419, 427, 438, 446, 453, 465, 491, 510, 518, 553, 616, 634, 635, 644, 670, 671, 679, 735, 787, 806, 833, 850, 852, 894, 913, 950, 958, 976, 1021, 1122, 1148, 1174, 1175, 1210, 1236, 1239, 1300, 1337, 1452, 1453, 1454, 1489, 1580, 1634, 1671, 1679, 1697, 1912, 1938, 1957, 1965, 2039, 2110, 2166, 2183, 2299, 2426, 2660, 3020, 3172, 3452, 3659, 3685, 3964, 4306, 4369, 4388, 4703, 4775, 4882, 4982, 5279, 5305, 5306, 5818, 8042

7

7, 15, 23, 39, 47, 53, 55, 61, 69, 88, 91, 97, 99, 113, 118, 121, 127, 128, 129, 135, 140, 143, 153, 161, 170, 183, 191, 193, 200, 205, 213, 215, 227, 235, 243, 257, 258, 265, 273, 289, 307, 319, 338, 341, 347, 349, 351, 360, 363, 372, 378, 380, 390, 393, 394, 421, 422, 425, 443, 451, 455, 465, 486, 487, 495, 499, 508, 516, 525, 539, 561, 569, 591, 613, 635, 643, 664, 675, 705, 715, 736, 737, 749, 758, 770, 778, 779, 788, 789, 811, 841, 863, 906, 915, 936, 967, 989, 990, 1009, 1039, 1042, 1061, 1083, 1113, 1115, 1126, 1135, 1156, 1187, 1209, 1221, 1262, 1292, 1304, 1327, 1334, 1335, 1357, 1365, 1379, 1407, 1410, 1437, 1440, 1441, 1511, 1512, 1524, 1534, 1577, 1607, 1660, 1690, 1691, 1755, 1808, 1827, 1831, 1849, 1901, 1922, 1923, 1932, 1944, 1975, 1985, 2005, 2036, 2058, 2153, 2176, 2206, 2247, 2257, 2259, 2286, 2287, 2300, 2320, 2373, 2383, 2395, 2426, 2456, 2521, 2531, 2627, 2751, 2763, 3131, 3149, 3223, 3381, 3476, 3651, 3695, 3927, 4063, 4314, 4461, 5481, 6781, 7105

8

7, 16, 25, 34, 43, 52, 59, 61, 68, 77, 93, 99, 102, 109, 111, 118, 127, 133, 136, 143, 144, 145, 152, 158, 161, 173, 182, 192, 194, 207, 216, 218, 219, 226, 232, 241, 243, 244, 257, 266, 268, 275, 292, 293, 301, 310, 328, 348, 349, 362, 384, 387, 394, 396, 398, 399, 409, 412, 422, 423, 429, 431, 439, 443, 446, 447, 448, 479, 480, 483, 513, 514, 517, 529, 544, 553, 554, 563, 567, 578, 587, 588, 597, 601, 613, 638, 647, 672, 674, 688, 697, 699, 722, 723, 729, 731, 733, 756, 769, 793, 803, 814, 818, 838, 839, 843, 852, 863, 864, 874, 877, 886, 897, 898, 899, 924, 958, 998, 1018, 1032, 1092, 1101, 1103, 1126, 1127, 1128, 1153, 1184, 1187, 1209, 1234, 1247, 1261, 1270, 1283, 1284, 1308, 1317, 1328, 1413, 1438, 1439, 1463, 1472, 1473, 1512, 1521, 1523, 1547, 1556, 1557, 1572, 1604, 1607, 1641, 1642, 1668, 1678, 1723, 1724, 1749, 1762, 1833, 1883, 1917, 1928, 1931, 2062, 2109, 2168, 2182, 2192, 2193, 2203, 2204, 2217, 2287, 2417, 2493, 2574, 2577, 2609, 2728, 2729, 2769, 2887, 2897, 2899, 3653, 3967, 4099, 4133, 4218, 4482, 4924, 5332, 5453, 5703, 6543, 7223, 7863

9

7, 17, 27, 37, 47, 57, 65, 67, 75, 77, 85, 93, 103, 110, 113, 121, 123, 131, 141, 148, 151, 159, 160, 161, 169, 176, 179, 193, 203, 214, 216, 217, 231, 241, 243, 244, 245, 252, 259, 262, 269, 271, 272, 273, 287, 297, 299, 300, 307, 317, 327, 328, 337, 347, 353, 367, 389, 390, 391, 405, 408, 425, 430, 433, 436, 441, 443, 445, 446, 453, 458, 461, 463, 472, 473, 474, 480, 482, 491, 492, 496, 499, 500, 501, 527, 537, 541, 565, 575, 579, 582, 593, 599, 610, 620, 631, 635, 637, 648, 658, 669, 673, 674, 680, 687, 690, 715, 725, 728, 745, 753, 755, 756, 763, 771, 773, 781, 783, 800, 810, 811, 820, 821, 848, 857, 889, 901, 913, 917, 940, 941, 945, 946, 956, 957, 968, 969, 980, 984, 994, 1006, 1007, 1008, 1037, 1071, 1075, 1081, 1120, 1143, 1158, 1237, 1238, 1264, 1265, 1266, 1294, 1329, 1332, 1357, 1385, 1399, 1400, 1405, 1415, 1425, 1440, 1441, 1453, 1468, 1478, 1491, 1571, 1581, 1587, 1615, 1626, 1653, 1709, 1710, 1737, 1747, 1748, 1751, 1765, 1801, 1804, 1842, 1843, 1845, 1873, 1884, 1935, 1936, 1941, 1952, 1964, 1978, 1979, 1990, 2115, 2121, 2153, 2168, 2169, 2221, 2247, 2251, 2395, 2435, 2451, 2462, 2463, 2474, 2475, 2490, 2531, 2569, 2579, 2747, 2800, 2854, 2893, 2895, 2937, 2972, 2974, 3065, 3243, 3257, 3338, 3915, 4073, 4201, 4257, 4549, 4569, 4607, 4743, 4837, 4966, 5049, 5729, 6129, 6636, 6724

10

7, 18, 29, 40, 51, 62, 71, 73, 82, 84, 93, 102, 113, 115, 121, 124, 133, 135, 144, 146, 155, 163, 166, 175, 176, 177, 186, 194, 197, 213, 224, 228, 236, 238, 239, 247, 255, 259, 266, 268, 269, 270, 278, 286, 289, 297, 299, 300, 301, 308, 317, 320, 328, 330, 331, 339, 350, 362, 373, 384, 390, 391, 406, 409, 421, 430, 431, 432, 448, 451, 452, 470, 476, 479, 482, 483, 488, 490, 492, 493, 501, 507, 510, 512, 522, 523, 524, 531, 533, 543, 544, 545, 549, 552, 553, 554, 584, 595, 599, 602, 626, 637, 641, 644, 645, 652, 657, 663, 676, 687, 694, 699, 703, 705, 718, 729, 735, 741, 745, 746, 747, 753, 761, 764, 783, 792, 795, 803, 806, 814, 825, 834, 836, 837, 838, 845, 847, 854, 856, 865, 867, 869, 887, 898, 907, 909, 940, 950, 951, 957, 985, 987, 994, 999, 1012, 1016, 1018, 1042, 1043, 1047, 1048, 1049, 1060, 1073, 1074, 1086, 1088, 1091, 1102, 1115, 1116, 1117, 1119, 1145, 1150, 1187, 1192, 1198, 1199, 1231, 1242, 1284, 1299, 1312, 1360, 1371, 1372, 1373, 1391, 1398, 1402, 1404, 1432, 1435, 1474, 1477, 1505, 1511, 1536, 1551, 1552, 1553, 1558, 1583, 1593, 1597, 1598, 1612, 1628, 1639, 1650, 1654, 1731, 1734, 1754, 1762, 1804, 1835, 1854, 1861, 1885, 1895, 1896, 1897, 1927, 1938, 1942, 1958, 1998, 2001, 2023, 2035, 2043, 2044, 2046, 2047, 2078, 2090, 2142, 2147, 2148, 2153, 2160, 2166, 2179, 2194, 2195, 2208, 2209, 2244, 2250, 2296, 2304, 2324, 2336, 2365, 2377, 2389, 2400, 2405, 2406, 2493, 2498, 2507, 2527, 2598, 2602, 2605, 2617, 2629, 2644, 2658, 2686, 2702, 2720, 2732, 2733, 2745, 2746, 2763, 2767, 2805, 2809, 2848, 2851, 2862, 3006, 3048, 3068, 3149, 3210, 3211, 3213, 3259, 3301, 3313, 3403, 3505, 3613, 3615, 3705, 3716, 3737, 3810, 3811, 3933, 4071, 4166, 4346, 4352, 4407

 

La tabella seguente riporta gli interi che richiedono 8 addendi, per piramidi centrate fino a 10 lati; se ve ne sono altri, sono maggiori di 1010 per le piramidi fino a 7 lati e di 3 • 1010 per quelle con 8 o più lati (M. Fiorentini, 2016).

Lati

Interi che richiedono 8 addendi

5

20, 360

6

15, 22, 50, 114, 167, 175, 186, 212, 231, 238, 303, 364, 420, 428, 454

7

8, 16, 24, 48, 56, 70, 100, 122, 130, 144, 162, 171, 192, 201, 202, 214, 236, 244, 266, 267, 274, 350, 381, 452, 496, 517, 526, 570, 644, 1366

8

8, 17, 26, 44, 53, 62, 69, 78, 103, 112, 128, 137, 146, 153, 162, 183, 193, 217, 227, 228, 229, 242, 267, 276, 278, 302, 303, 311, 397, 432, 433, 564, 598, 648, 732, 853, 887, 1318

9

8, 18, 28, 38, 48, 58, 68, 76, 86, 88, 104, 114, 124, 132, 142, 152, 162, 170, 171, 180, 204, 215, 242, 253, 254, 255, 256, 270, 298, 308, 310, 311, 338, 339, 348, 444, 447, 483, 484, 485, 502, 538, 576, 621, 632, 659, 670, 726, 784, 822, 995, 1479

10

8, 19, 30, 41, 52, 63, 74, 83, 85, 94, 96, 114, 125, 127, 136, 145, 156, 158, 167, 178, 187, 188, 189, 198, 225, 237, 240, 267, 271, 279, 280, 281, 282, 290, 298, 302, 329, 332, 340, 342, 343, 344, 351, 363, 374, 385, 433, 491, 494, 495, 513, 525, 534, 535, 536, 555, 556, 596, 638, 688, 700, 706, 730, 742, 804, 807, 846, 857, 868, 899, 910, 1061, 1103, 1403, 1640, 1939

 

La tabella seguente riporta gli interi che richiedono più di 8 addendi, per piramidi centrate fino a 10 lati; se ve ne sono altri, sono maggiori di 1010 per le piramidi fino a 7 lati e di 3 • 1010 per quelle con 8 o più lati (M. Fiorentini, 2016).

Lati

Interi che richiedono 9 addendi

Interi che richiedono 10 addendi

Interi che richiedono 11 addendi

Interi che richiedono 12 addendi

Interi che richiedono 13 addendi

6

23, 239

 

 

 

 

7

17, 25, 57, 131, 245, 275, 276

26

 

 

 

8

9, 18, 27, 54, 63, 79, 113, 138, 147, 163, 277, 312, 313

19, 28, 64, 148

29

 

 

9

9, 19, 29, 49, 59, 69, 87, 115, 125, 143, 153, 163, 181, 182, 309, 349, 350

10, 20, 30, 60, 70, 126, 154, 164

21, 31, 71, 165

32

 

10

9, 20, 31, 42, 53, 64, 75, 95, 97, 126, 137, 139, 157, 168, 170, 179, 199, 200, 201, 283, 341, 375, 386, 387, 537, 911

10, 21, 32, 54, 65, 76, 138, 169, 180, 182

11, 22, 33, 66, 77, 181

23, 34, 78

35

 

Per piramidi fino a 10 lati ho verificato che ogni intero positivo abbastanza grande può essere espresso come somma di numeri piramidali centrati distinti con numero di lati fissato; probabilmente vale per qualsiasi numero di lati, ma non credo sia stato dimostrato.

La tabella seguente riporta il numero di interi non rappresentabili come somma di numeri piramidali centrati differenti e il massimo, per piramidi fino a 10 lati.

Lati

Numero totale di interi non rappresentabili come somma di numeri piramidali centrati differenti

Massimo intero non rappresentabili come somma di numeri piramidali centrati differenti

3

504

2313

4

819

3255

5

1436

5863

6

2788

12758

7

2691

9299

8

3051

11088

9

3316

10212

10

3998

12543

 

La tabella seguente riporta i numeri inferiori a 100 non rappresentabili come somma di numeri piramidali centrati distinti, per piramidi fino a 10 lati.

Lati

Interi minori di 100 non rappresentabili come somma di numeri piramidali centrati differenti

3

2, 3, 4, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 36, 37, 38, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 68, 69, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98

4

2, 3, 4, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 48, 49, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 87, 88, 89, 90, 93, 94, 95, 96, 97, 98, 99

5

2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 56, 57, 58, 59, 60, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 79, 80, 81, 82, 83, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

6

2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 66, 67, 68, 69, 70, 71, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 93, 94, 95, 96, 97, 98

7

2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 76, 77, 78, 79, 80, 81, 82, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

8

2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 86, 87, 88, 89, 90, 91, 92, 93, 96, 97, 98, 99

9

2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 41, 42, 43, 44, 45, 46, 47, 48, 49, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 96, 97, 98, 99

10

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

 

Contattami

Potete contattarmi al seguente indirizzo bitman[at]bitman.name per suggerimenti o segnalazioni d'errori relativi a questo articolo.