User talk:Matthijs Coster

Multiplicative dependent consecutive numbers

We define the set ${Exp}_{S}=\{\prod m_{i}^{z_{i}}|m_{i}\in S,z_{i}\in Z\}$, where $S$ is a finite set. Consider the set $S_{N}=\{N,N+1,N+2,\dots ,N+k\}$ and the related set $Exp_{S_{N}}$. We wonder how large $k$ must be chosen such that there exists a dependency, in other words there are $k_{1},k_{2},\dots ,k_{t}$ and non-zero integers $z_{1},z_{2},\dots ,z_{t}$ such that $k_{1}^{z_{1}}k_{2}^{z_{2}}\dots k_{t}^{z_{t}}=1$. We search for the smallest $k$. We define $S$ multiplicative dependent if in $Exp_{S}$ there exists a dependency.

Examples

(1) $N=126$. The set $\{126,128,130,132,135,140,143\}$ is multiplicative dependent since $126^{14}\cdot 128^{-13}\cdot 130^{35}\cdot 132^{35}\cdot 135^{-21}\cdot 140^{-14}\cdot 143^{-35}=1$.

(2) $N=55$. The set $\{55,56,60,63,64,66\}$ is multiplicative dependent since $55\cdot 56^{-1}\cdot 60^{-1}\cdot 63\cdot 64\cdot 66^{-1}=1$.

Sequences

We distinguish two different types of sets: exponential sets and linear sets. Example (1) is an example of an exponential set. Typically the sum of the positive exponents is 1 (sometimes 2 or more) larger than the sum of the negative exponents. Example (2) is an example of an linear set. Typically the sum of the positive exponents is equal to the sum of the negative exponents. I calculated all multiplicative dependent sets up to 10.000. I created 7 sequences:

• First numbers: 2, 3, 4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 33, 35, 36, 40, 42, 45, 48, 49, 50, 54, 55, 56, 60, 63, 64, 65, 66, 72, 75, 77, 78, 80, 81, 84, 88, 90, 91, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 117, 120, 121, 125, 126, 128, 130, 135, 136, 140, 143, 144, 147, 150, 154, 156, 160, 162, 165, 168, 169, 170, 171, 175, 180, 182, 187, 189, 190, 192, 195, 196, 198, 200, 204, 209, 210, 216, 220, 221, 224, 225, 228, 231, 240, 243, 245, 247, 250, 252, 255, 256, 260, 264, 270, 272, 275, 276, 280, 285, 288, 289, 290, 294, 299, 300, 304, 308, 312, 315, 320, 323, 324, 325, 330, 336, 338, 340, 342, 343, 345, 350, 351, 352, 357, 360, 361, 363, 364, 368, 375, 380, 384, 385, 390, 391, 392, 396, 399, 400, 403, 406, 408, 414, 416, 420, 425, 429, 432, 437, 440, 441, 442, 448, 450, 455, 456, 459, 460, 462, 464, 468, 480, 484, 486, 490, 494, 504, 506, 507, 512, 520, 522, 525, 527, 528, 529, 540, 544, 546, 550, 551, 552, 560, 561, 567, 572, 576, 578, 580, 588, 589, 594, 595, 598, 600, 605, 608, 609, 621, 624, 625, 627, 630, 637, 640, 646, 648, 650, 651, 663, 665, 667, 672, 675, 676, 680, 682, 686, 696, 700, 702, 704, 715, 720, 722, 725, 726, 728, 729, 735, 736, 744, 748, 750, 756, 760, 765, 768, 770, 775, 777, 782, 783, 784, 800, 805, 810, 812, 816, 819, 825, 832, 840, 841, 845, 847, 850, 851, 858, 864, 868, 875, 880, 882, 891, 896, 897, 899, 902, 903, 912, 918, 924, 928, 930, 931, 945, 950, 952, 957, 960, 962, 968, 972, 975, 980, 986, 990, 992, 1000, 1001, ...

• Differences between first and last number: 2, 3, 4, 3, 4, 6, 6, 6, 5, 8, 7, 7, 7, 6, 7, 8, 8, 10, 10, 11, 10, 12, 9, 8, 9, 8, 11, 13, 10, 11, 14, 12, 12, 13, 13, 14, 9, 13, 13, 13, 16, 17, 15, 12, 14, 14, 12, 12, 13, 17, 17, 16, 16, 17, 16, 16, 13, 12, 14, 15, 17, 16, 17, 15, 17, 16, 17, 18, 18, 18, 15, 19, 16, 18, 17, 19, 20, 20, 21, 20, 16, 18, 17, 19, 19, 18, 21, 24, 23, 24, 21, 19, 21, 22, 20, 21, 19, 20, 19, 19, 12, 13, 21, 23, 22, 21, 20, 24, 25, 22, 18, 22, 22, 24, 24, 21, 20, 26, 29, 26, 23, 24, 21, 22, 24, 23, 22, 20, 26, 26, 22, 21, 22, 21, 21, 21, 23, 24, 24, 26, 23, 24, 24, 27, 27, 24, 21, 19, 21, 29, 26, 27, 28, 29, 30, 32, 31, 29, 32, 27, 26, 28, 25, 26, 27, 23, 22, 27, 33, 28, 30, 28, 28, 27, 30, 31, 30, 27, 20, 20, 26, 23, 26, 21, 26, 32, 28, 26, 29, 27, 31, 32, 32, 27, 31, 30, 28, 29, 33, 28, 33, 28, 26, 24, 31, 32, 28, 31, 30, 30, 29, 30, 32, 30, 31, 23, 24, 25, 36, 35, 30, 32, 30, 32, 32, 33, 23, 31, 33, 30, 29, 37, 34, 33, 34, 29, 28, 27, 31, 26, 30, 32, 31, 33, 32, 36, 33, 34, 31, 34, 33, 28, 38, 40, 38, 40, 37, 37, 34, 42, 44, 32, 32, 30, 33, 31, 31, 30, 26, 24, 34, 35, 35, 34, 37, 33, 32, 31, 25, 38, 38, 33, 35, 38, 37, 41, 43, 38, 39, 36, 33, 38, 38, 27, 36, 36, 33, 32, 37, 33, 36, 39, 40, 37, 34, 37, 35, 39, 36, 33, 36, 38, 34, 32, 32, 35, 38, 42, 40, 38, 31, 33, 36, 34, 37, 38, 40, 33, 42, 40, 42, 43, 41, 40, 32, 34, 29, 42, 39, 38, 38, 41, 30, 37, 39, 38, 37, 34, 35, 40, 43, 39, 40, 39, 42, 35, 41, 39, 45, 40, 38, 36, 29, 43, 39, 38, 37, 38, 44, 45, 43, 41, 40, 42, 42, 39, 28, 34, 42, 40, 39, 40, 47, 46, 46, 49, 48, 37, 31, 33, 35, 30, 35, 37, 33, 44, 47, 47, 46, 49, 47, 46, 43, 38, 45, 48, 44, 44, 47, 43, 38, 41, 41, 48, 52, 45, 45, 44, 40, 43, 45, 47, 41, 40, 44, 47, 33, 45, 43, 42, 47, 51, 38, 48, 46, 48, 52, 52, 47, 45, 50, 47, 44, 47, 37, 31, 36, 41, 45, 39, 40, 31, 45, 44, 44, 42, 40, 42, 48, 35, 39, 40, 43, 49, 47, 49, 48, 47, 53, 54, 49, 46, 51, 53, 49, 48, 50, 48, 48, 47, 45, 45, 41, ...

• Size (How many numbers are involved?): 2, 3, 2, 3, 3, 4, 4, 3, 4, 3, 4, 4, 4, 4, 4, 5, 5, 4, 5, 6, 5, 4, 5, 5, 4, 4, 5, 5, 4, 6, 5, 5, 5, 6, 7, 6, 4, 6, 6, 6, 4, 4, 5, 5, 6, 6, 5, 6, 6, 6, 6, 6, 6, 5, 6, 5, 5, 5, 5, 5, 7, 6, 7, 5, 6, 7, 7, 4, 6, 6, 7, 6, 6, 6, 7, 8, 8, 8, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 7, 7, 7, 7, 6, 8, 8, 8, 6, 6, 8, 6, 5, 5, 6, 7, 5, 7, 6, 7, 8, 8, 8, 8, 7, 8, 6, 6, 6, 7, 7, 6, 8, 6, 7, 7, 6, 6, 7, 8, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 7, 8, 5, 7, 7, 9, 8, 6, 7, 6, 8, 7, 9, 7, 8, 9, 5, 8, 7, 9, 9, 8, 8, 8, 7, 7, 9, 8, 7, 8, 8, 8, 9, 9, 8, 7, 8, 9, 7, 5, 6, 5, 7, 6, 6, 9, 8, 7, 8, 8, 7, 8, 7, 8, 7, 9, 8, 8, 8, 9, 8, 6, 8, 9, 8, 8, 8, 7, 9, 9, 9, 8, 7, 8, 9, 8, 7, 6, 6, 9, 8, 11, 11, 8, 6, 9, 10, 8, 8, 7, 6, 7, 9, 9, 9, 7, 8, 7, 6, 7, 8, 7, 8, 7, 7, 7, 8, 8, 8, 8, 8, 10, 10, 9, 8, 8, 8, 8, 8, 8, 7, 7, 8, 9, 8, 8, 8, 8, 9, 9, 9, 9, 8, 7, 8, 11, 8, 6, 8, 6, 8, 8, 9, 10, 12, 11, 11, 10, 9, 9, 9, 10, 9, 8, 8, 9, 10, 9, 6, 8, 7, 7, 7, 9, 10, 8, 7, 8, 8, 7, 10, 7, 7, 10, 8, 9, 9, 11, 10, 8, 7, 9, 7, 8, 8, 11, 9, 9, 9, 9, 11, 11, 11, 11, 8, 6, 8, 9, 9, 9, 9, 8, 9, 8, 11, 9, 9, 6, 8, 7, 9, 11, 9, 8, 8, 10, 10, 11, 9, 8, 8, 10, 10, 9, 10, 10, 10, 9, 11, ...

• All numbers involved: 2, 4, 3, 4, 6, 4, 8, 6, 8, 9, 8, 9, 12, 9, 10, 12, 15, 10, 12, 15, 16, 12, 16, 18, 15, 16, 18, 20, 16, 18, 24, 18, 20, 24, 25, 20, 24, 25, 27, 21, 24, 27, 28, 24, 25, 27, 30, 25, 27, 30, 32, 27, 28, 30, 32, 35, 28, 30, 32, 35, 36, 30, 32, 36, 40, 32, 35, 36, 40, 42, 33, 35, 36, 40, 42, 44, 35, 36, 40, 42, 45, 36, 40, 45, 48, 40, 42, 45, 48, 49, 42, 45, 48, 49, 50, 45, 48, 50, 54, 48, 49, 54, 56, 49, 50, 54, 56, 60, 50, 54, 56, 60, 63, 54, 56, 63, 64, 55, 56, 60, 63, 64, 66, 56, 60, 63, 64, 70, 60, 63, 64, 70, 72, 63, 64, 70, 72, 75, 64, 66, 70, 72, 75, 77, 65, 66, 70, 72, 77, 78, 66, 70, 72, 75, 77, 80, 72, 75, 80, 81, 75, 77, 80, 81, 84, 88, 77, 80, 81, 84, 88, 90, 78, 80, 81, 84, 90, 91, 80, 81, 90, 96, 81, 84, 96, 98, 84, 88, 96, 98, 99, 88, 90, 96, 99, 100, 90, 91, 96, 98, 100, 104, 91, 96, 98, 100, 104, 105, 96, 98, 100, 105, 108, 98, 99, 100, 105, 108, 110, 99, 100, 105, 108, 110, 112, 100, 104, 105, 108, 112, 117, ... This must be read as {2, 4}, {3, 4, 6}, {4, 8}, {6, 8, 9}, {8, 9, 12}, {9, 10, 12, 15}, {10, 12, 15, 16}, {12, 16, 18}, {15, 16, 18, 20}, ...

• The exponential case: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 2, ... Here we take the difference between the sum of the positive exponents and the sum of the negative exponents. The first 2 in this sequence appears for the set {125, 126, 128, 135, 140}, here the corresponding exponents are {35, 63, 9, -42, -63}.

• The linear case: 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 5, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, 4, 6, 3, 3, 3, 3, 4, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 5, 3, 3, 4, 4, 3, 3, 4, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 4, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 4, 4, 3, 3, 4, 4, 3, 3, 4, 3, 3, 5, 4, 3, 3, 5, 3, 3, 3, 3, 3, 17, 3, ... Here the sum of the positive exponents equals to the sum of the negative exponents. Therefore we only consider only the sum of the positive exponents. The number 17 appears for the set: {2394, 2397, 2400, 2401, 2405, 2420, 2430, 2431, 2432, 2442, 2444}, here the corresponding exponents are {-4, 2, -5, 1, 4, 3, 3, -2, 4, -4, -2}.

• Combination of exponential and linear case: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -3, 1, 1, 1, 1, 1, 1, 1, 1, 1, -3, 1, 1, 1, 1, -3, 1, 1, 1, 1, -3, 1, 1, 1, 1, 1, 1, -3, 1, 1, 1, -3, 1, 1, 1, 1, 1, -3, 1, 1, 2, 1, -3, 1, 1, -3, 1, 1, 1, 1, 1, -4, 1, 1, -3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -3, 1, 1, 1, 1, 1, 1, -3, 1, 1, -3, -5, 1, 1, -4, 1, 1, 1, 1, 1, 1, -3, 1, -3, 1, -3, -3, 1, -3, -4, 1, 1, 1, 1, -3, 1, -3, 1, 1, -3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, -3, 1, 1, 1, 1, -3, 1, 3, 2, 2, 1, -3, 1, 1, -3, 1, -3, 1, 1, 1, -3, -3, 1, 1, -3, -3, 1, -3, 1, -4, -3, 1, 1, -3, -3, 1, 1, -3, 1, 1, 2, 1, -3, 1, 1, 1, 1, -3, 1, -4, 1, -3, -3, 1, -3, 1, 1, -3, 1, 1, 1, -3, 1, 1, 1, 1, 1, -3, 1, 1, -3, -4, 1, 1, 1, -3, 1, 1, 1, -3, 1, -3, 1, 1, -4, -3, 1, 1, -4, 1, 1, 1, 1, 1, 1, 1, -3, 1, 1, -3, 1, -3, 1, 1, -5, 1, 1, 1, 1, 1, 1, 1, -3, 1, 1, -3, -3, 1, 1, 1, 1, -3, -3, 1, 1, -3, -4, -3, 1, 1, 1, 1, -3, -3, -4, 1, 1, 1, 1, 3, 1, 1, -3, 1, 1, -3, 1, 1, 1, -4, -3, -3, 1, -3, -4, -6, 1, -3, 1, 3, 1, 1, -3, 1, -3, -3, 1, -4, 1, 1, 1, -3, 1, -4, 1, 1, 3, -3, 1, 1, 1, 2, 1, -3, -3, -3, 1, -3, 1, 1, 1, -3, -3, 1, 1, 1, -3, 1, -3, 1, -3, 1, -3, 1, 1, 1, -3, 1, 1, 1, -3, -3, 1, 1, -4, 1, 1, 1, 1, -3, -4, -4, 1, 1, -3, -3, -3, -3, 1, -3, -3, 1, 1, 1, 1, -3, 1, -3, 1, 1, ... Here the positive numbers correspond to the exponential case, while the negative numbers correspond to the linear case.

Multiplicative dependent consecutive odd integers

We now consider the set $\{N,N+2,N+4,N+6,\dots ,N+2k\}$, such that there is an multiplicative dependency. We find the following sequences:

• First numbers: 3, 5, 7, 9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 55, 63, 65, 75, 77, 81, 85, 91, 95, 99, 117, 119, 121, 125, 133, 135, 143, 147, 153, 161, 165, 169, 171, 175, 187, 189, 195, 207, 209, 221, 225, 231, 243, 245, 247, 253, 255, 261, 273, 275, 279, 285, 299, 315, 323, 325, 343, 345, 361, 363, 375, 377, 385, 391, 403, 407, 425, 429, 435, 441, 455, 459, 475, 481, 483, 493, 495, 507, 513, 525, 529, 533, 539, 561, 567, 575, 585, 589, 595, 605, 609, 615, 625, 627, 629, 637, 645, 651, 665, 667, 675, 693, 697, 703, 713, 715, 725, 729, 731, 735, 759, 765, 775, 777, 779, 783, 819, 825, 833, 837, 845, 847, 851, 855, 867, 875, 891, 893, 897, 925, 931, 945, 957, 961, 969, 975, 987, 989, 999, 1001, 1023, 1025, 1029, 1035, 1045, 1053, 1071, 1073, 1081, 1089, 1105, 1107, 1113, 1125, 1147, 1175, 1183, 1189, 1197, 1215, 1225, 1235, 1247, 1269, 1275, 1287, 1295, 1309, 1325, 1331, 1333, 1353, 1363, 1365, 1375, 1377, 1419, 1421, 1425, 1431, 1435, 1445, 1449, 1457, 1463, 1485, 1495, 1505, 1519, 1521, 1537, 1551, 1573, 1591, 1593, 1595, 1599, 1615, 1617, 1625, 1643, 1645, 1647, 1677, 1683, 1701, 1705, 1711, 1715, 1729, 1739, 1749, 1755, 1763, 1771, 1785, 1805, 1815, 1827, 1833, 1849, 1855, 1859, 1863, 1885, 1891, 1911, 1925, 1927, 1947, 1953, 1961, 1995, 2009, 2021, 2023, 2025, 2035, 2057, 2065, 2067, 2079, 2107, 2115, 2125, 2145, 2175, 2187, 2197, 2205, 2211, 2233, 2255, 2261, 2295, 2299, 2301, 2303, 2325, 2331, 2345, 2349, 2365, 2387, 2401, 2431, 2457, 2465, 2475, 2491, 2499, 2511, 2523, 2527, 2535, 2541, 2565, 2583, 2597, 2601, 2623, 2625, 2635, 2645, 2655, 2673, 2703, 2755, 2793, 2821, 2835, 2849, 2867, 2873, 2875, 2883, 2891, 2907, 2911, 2915, 2923, 2925, 2945, 2961, 2975, 2989, 2997, 3003, 3025, 3059, ...

• Differences between first and last number: 6, 10, 14, 16, 12, 14, 20, 22, 22, 28, 26, 30, 28, 26, 28, 34, 30, 40, 38, 36, 34, 38, 36, 26, 34, 44, 44, 38, 40, 44, 42, 42, 46, 44, 52, 54, 56, 56, 56, 52, 46, 46, 52, 50, 54, 46, 52, 52, 62, 64, 62, 52, 66, 64, 60, 52, 46, 52, 60, 48, 54, 44, 72, 62, 64, 70, 68, 62, 74, 68, 66, 72, 72, 72, 70, 64, 74, 78, 74, 80, 78, 76, 70, 76, 82, 82, 64, 70, 76, 78, 76, 72, 70, 84, 82, 78, 88, 96, 92, 86, 84, 76, 92, 100, 84, 82, 80, 92, 102, 94, 96, 102, 102, 82, 80, 72, 74, 82, 84, 56, 66, 66, 88, 86, 88, 92, 90, 94, 94, 96, 96, 102, 76, 92, 84, 96, 110, 104, 100, 94, 94, 86, 88, 82, 82, 96, 92, 102, 102, 90, 100, 94, 100, 110, 112, 108, 100, 88, 94, 112, 116, 112, 96, 98, 118, 116, 96, 94, 88, 82, 110, 106, 112, 112, 104, 100, 114, 110, 128, 98, 100, 112, 108, 112, 106, 124, 118, 118, 102, 104, 110, 98, 104, 106, 94, 108, 92, 118, 120, 130, 124, 132, 130, 120, 118, 122, 94, 102, 104, 110, 118, 118, 120, 120, 114, 120, 118, 114, 102, 106, 110, 100, 102, 112, 134, 136, 138, 128, 132, 114, 132, 138, 120, 126, 130, 114, 106, 104, 150, 158, 150, 130, 128, 130, 126, 102, 118, 136, 130, 102, 108, 106, 132, 134, 142, 130, 126, 106, 132, 150, 154, 140, 144, 134, 142, 134, 124, 134, 110, 108, 132, 126, 134, 136, 128, 142, 146, 144, 150, 130, 114, 106, 136, 122, 130, 138, 138, 140, 132, 106, 116, 114, 124, 154, 154, 142, 142, 146, 142, 164, 162, 164, 166, 164, 180, 166, 164, 152, 146, 148, 146, 132, 126, 136, 126, 134, 114, 146, 140, 152, 136, 140, 142, 144, 130, 148, 168, 186, 178, 168, 176, 162, 156, 174, 160, 186, 176, 174, 172, 170, 166, 158, 146, 126, 144, 146, 142, 126, 160, 136, 158, 156, 146, 150, 158, 142, 148, 168, 138, 140, 146, 140, 140, 136, 170, 160, 134, 128, 142, 154, 156, 158, 172, 164, 166, 176, 174, 174, 162, 170, 154, 186, 172, 188, 186, 172, 174, 176, 174, 130, 112, 138, 144, 182, 170, ...

• Size (How many numbers are involved?): 2, 3, 3, 3, 3, 4, 3, 4, 5, 4, 5, 4, 5, 5, 5, 6, 5, 5, 6, 7, 6, 6, 4, 5, 5, 4, 5, 7, 6, 7, 6, 7, 6, 7, 8, 8, 7, 7, 4, 7, 6, 8, 6, 5, 7, 7, 8, 8, 8, 6, 7, 8, 7, 8, 8, 6, 7, 7, 6, 7, 8, 6, 6, 10, 10, 9, 7, 7, 8, 8, 10, 9, 7, 8, 8, 7, 8, 9, 8, 6, 6, 9, 7, 7, 8, 7, 7, 6, 9, 9, 9, 10, 8, 8, 9, 9, 9, 12, 9, 11, 7, 7, 7, 7, 10, 12, 11, 7, 11, 9, 10, 10, 8, 9, 8, 8, 9, 9, 8, 7, 6, 7, 10, 9, 7, 10, 7, 8, 9, 10, 13, 13, 11, 10, 8, 8, 9, 10, 9, 9, 11, 10, 8, 9, 9, 8, 9, 11, 10, 11, 12, 8, 10, 9, 9, 10, 8, 9, 7, 8, 11, 9, 10, 10, 13, 11, 9, 8, 7, 7, 8, 7, 10, 11, 12, 12, 10, 8, 9, 13, 12, 13, 13, 11, 11, 10, 9, 9, 9, 11, 11, 9, 8, 11, 9, 9, 12, 11, 10, 9, 10, 9, 10, 12, 9, 10, 11, 10, 9, 9, 11, 10, 12, 11, 10, 7, 7, 9, 10, 8, 7, 9, 9, 11, 10, 9, 10, 12, 12, 10, 11, 12, 11, 10, 10, 9, 9, 8, 9, 11, 13, 13, 14, 10, 8, 7, 8, 9, 7, 9, 7, 8, 10, 11, 10, 13, 13, 11, 13, 14, 12, 12, 11, 12, 12, 11, 10, 8, 8, 9, 10, 9, 9, 7, 9, 8, 8, 9, 10, 10, 11, 11, 10, 12, 11, 13, 11, 11, 10, 11, 12, 11, 10, 10, 12, 11, 12, 14, 11, 11, 11, 14, 12, 9, 10, 12, 9, 10, 11, 11, 10, 8, 9, 8, 8, 11, 8, 10, 10, 9, 9, 9, 12, 11, 10, 10, 11, 12, 11, 10, 10, 12, 12, 12, 12, 12, 11, 13, 16, 17, 17, 15, 15, 13, 14, 12, 13, 12, 11, 12, 11, 10, 11, 12, 11, 12, 10, 9, 10, 12, 11, 12, 11, 10, 12, 12, 14, 11, 11, 12, 12, 12, 11, 9, 11, 9, 13, 11, 12, 14, 9, 14, 13, 14, 14, 15, 14, 12, 15, 10, 10, 8, 9, 11, 15, 11, 13, 11, 11, 10, 10, 10, 9, 13, 12, 11, 12, 15, 11, 12, 13, 13, 13, 11, 10, 12, 12, 10, 10, 12, 16, 13, 16, 12, 15, 13, 11, 14, 15, 9, 9, 9, 11, 10, 11, 12, 12, 10, 11, 11, 13, 16, 15, 14, 13, 15, 12, 12, 14, 15, 13, 11, 15, 17, 16, 15, 15, 15, 14, 11, 11, 13, 11, 10, 11, 11, 11, 11, 13, 14, 11, 12, 11, 11, 15, 15, 14, 13, 12, 15, 14, 14, 14, 15, 14, 10, 12, 13, 13, 13, 12, 12, 11, 13, 14, 15, 14, 11, 10, 11, 11, 13, 12, 11, 11, 12, 15, ...

• All numbers involved: 3, 9, 5, 9, 15, 7, 9, 21, 9, 15, 25, 15, 25, 27, 21, 25, 27, 35, 25, 27, 45, 27, 35, 45, 49, 33, 35, 45, 49, 55, 35, 45, 49, 63, 39, 45, 49, 63, 65, 45, 49, 63, 75, 49, 55, 63, 75, 77, 55, 63, 75, 77, 81, 63, 65, 75, 81, 91, 65, 75, 77, 81, 91, 99, 75, 77, 81, 99, 105, 77, 91, 99, 117, 81, 85, 91, 105, 117, 119, 85, 91, 99, 105, 117, 119, 121, 91, 99, 105, 117, 121, 125, 95, 99, 105, 121, 125, 133, 99, 121, 125, 135, 117, 121, 125, 135, 143, 119, 125, 135, 147, 153, 121, 125, 135, 165, 125, 135, 143, 165, 169, 133, 135, 143, 147, 165, 169, 171, 135, 143, 147, 165, 169, 175, 143, 147, 153, 165, 169, 175, 187, 147, 153, 165, 175, 187, 189, 153, 165, 169, 175, 187, 189, 195, 161, 169, 175, 195, 207, 165, 169, 171, 175, 189, 195, 209, 169, 171, 175, 187, 189, 195, 209, 221, 171, 175, 187, 189, 195, 209, 221, 225, 175, 187, 189, 195, 221, 225, 231, 187, 189, 195, 221, 225, 231, 243, 189, 225, 243, 245, 195, 209, 225, 231, 243, 245, 247, 207, 225, 231, 243, 245, 253, 209, 221, 225, 231, 243, 245, 247, 255, 221, 225, 243, 245, 255, 273, 225, 231, 243, 245, 275, 231, 243, 245, 247, 273, 275, 285, 243, 245, 247, 255, 273, 285, 289, 245, 247, 255, 273, 275, 285, 289, 297, 247, 253, 255, 275, 285, 289, 297, 299, 253, 255, 273, 275, 289, 297, 299, 315, 255, 261, 275, 289, 319, 261, 275, 285, 289, 319, 323, 273, 275, 285, 289, 297, 315, 323, 325, 275, 279, 285, 289, 323, 341, 279, 285, 289, 297, 315, 323, 341, 343, 285, 289, 299, 315, 323, 325, 343, 345, 299, 315, 325, 343, 345, 351, 315, 323, 325, 343, 351, 357, 361, 323, 325, 343, 351, 357, 361, 375, 325, 343, 351, 363, 375, 385, 343, ...

• The exponential case: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 2, 5, 1, ... Here we take the difference between the sum of the positive exponents and the sum of the negative exponents.

• The linear case: 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 4, 3, 3, 4, 4, 5, 4, 7, 4, 5, 3, 4, 4, 8, 5, 4, 3, 3, 3, 5, 4, 3, 7, 3, 5, 3, 4, 3, 4, 3, 4, 3, 4, 6, 3, 4, 4, 6, 5, 5, 3, 4, 4, 4, 6, 7, 4, 3, 4, 4, 4, 6, 5, 4, 3, 4, 3, 3, 3, 5, 4, 5, 5, 8, 4, 3, 4, 4, 4, 5, 4, 4, 3, 5, 3, 3, 3, 4, 3, 4, 4, 4, 4, 3, 4, 4, 3, 3, 3, 4, 6, 3, 4, 3, 3, 4, 7, 5, 7, 3, 4, 4, 4, 4, 3, 4, 5, 3, 4, 3, 4, 4, 3, 4, 5, 5, 3, 7, 4, 3, 4, 6, 8, 8, 3, 5, 3, 3, 3, 3, 5, 4, 12, 4, 3, 4, 3, 5, 3, 4, 4, 7, 3, 3, 3, 7, 5, 5, 3, 3, 3, 7, 9, 3, 5, 3, 3, 3, 3, 3, 3, 5, 4, 3, 4, 8, 3, 5, 4, 7, 7, 3, 6, 6, 17, 3, 4, 8, ... Here the sum of the positive exponents equals to the sum of the negative exponents. Therefore we only consider only the sum of the positive exponents.

• Combination of exponential and linear case: 1, 1, 1, -2, 1, 1, 1, 1, 1, -2, 1, 1, 1, 1, 1, 1, 1, -2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -3, -4, 1, -4, 1, 1, 1, 1, 1, 1, -4, 1, -4, 1, -5, 1, 1, 1, 1, -4, 1, -3, 1, -3, 1, 1, 1, 1, -4, 1, 1, -4, -5, 1, 1, -4, 1, 1, 1, -7, 1, 1, 1, 1, 1, -4, 1, 1, 1, 1, -5, 1, -3, 1, 1, 1, -4, 1, 1, 1, 1, 1, 1, 1, 1, -4, 1, 1, 1, 1, 1, 1, 1, -8, 1, 2, 2, 1, -5, 1, 1, 1, 1, -4, 1, -3, 1, 1, 1, 1, -3, 1, 1, 1, 1, -3, 1, 1, -5, 1, -4, 1, -3, 1, 1, 2, -7, 1, 1, 1, 1, 1, -3, -5, -3, 1, 1, -4, 1, 1, 1, 1, -3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -4, 1, 1, 1, 1, 1, 1, 1, -3, -4, -3, 1, -4, 1, 1, 1, -6, 1, -3, 1, 1, 1, 1, -4, 1, -4, 1, 1, -6, 1, -5, 1, -5, 1, 1, 1, -3, -4, 1, 1, 2, 1, -4, -4, -6, 1, 1, -7, -4, -3, 1, -4, 1, -4, -4, 1, 1, -6, -5, 1, 1, -4, 1, -3, 1, -4, -3, 1, 1, -3, -3, -5, -4, -5, -5, -8, 1, 1, 1, -4, 1, 1, -3, 1, -4, 1, 1, 1, 1, -4, 1, 1, 1, 1, 1, 1, 1, 1, -4, -5, -4, 1, -4, 1, -3, -5, -3, -3, -3, 1, 1, 1, 1, 1, 1, 1, 1, 1, -4, 1, -3, 1, 1, 1, 1, 3, 1, 1, -4, -4, 1, 3, -4, 1, -4, 1, 1, -3, -4, 1, -4, 1, 1, -3, 1, 1, -3, 1, 1, -3, -4, 1, 1, -6, 1, 1, 1, 1, 1, -3, 1, 1, 1, -4, 1, -3, -3, 1, -4, -7, 1, 2, 1, 1, -5, -7, -3, 1, -4, -4, 1, 1, -4, -4, 1, 1, 3, 1, 1, -3, -4, 1, -5, -3, 1, 1, ... Here the positive numbers correspond to the exponential case, while the negative numbers correspond to the linear case.

Some final remarks

There can be written much more about multiplicative dependent consecutive numbers.

• How to construct such a set?

>(1) factor integers N, N+1, N+2, ..., N+k until the number of primes is one larger 
     than the number of integers. 
>(2) remove the integers which appear only once. 
>(3) If the number of primes is two (or more) larger than the number of integers, 
     remove the largest integer from the set, go to (2). 
>(4) create an incidence matrix of integers and primes, and use this matrix as set
     of vectors (for each prime a vector).
>(5) find a vector orthogonal to all these vecors.

• The difference of the largest and smallest number in sets is about $4{\sqrt[{3}]{N}}$. Probably this can be proven easily using the theory of smooth numbers.

• {245, 247, 250, 256, 260, 266} is probably the only set which do not contain the prime 3, but contains larger primes.

• multiplicative dependent sets starting with 2, 3, 4, 6, 8, 12, 16, 21, 48, 54, 81, 84, 102, 187, 288, 561, 777, 992, 1056, 2352, 2496, 2737, ... do not contain the prime 5.

• linear sets of size 6 have always (as far as I could see) the related exponents {1, -1, -1, 1, 1, -1}. (See example (2)).

----Matthijs Coster 23:05, 10 March 2010 (UTC)