Recently, I noticed that the numbers from 27872 to 27876 had four or more prime factors with multiplicity. Specifically:$$ \begin{align} 27872 &= 2^5 \times 13 \times 67 \\ 27873 &= 3^2 \times 19 \times 163 \\ 27874 &= 2 \times 7 \times 11 \times 181 \\ 27875 &= 5^3 \times 223 \\ 27876 &= 2^2 \times 3 \times 23 \times 101 \end{align} $$This got me wondering how often these runs of five numbers with four or more factors occur. It turns out not often. In the range up to 40000, there are only 11 such numbers:$$12122, 14748, 17574, 19940, 22382, 27872, 28592, 29886, 35072, 35124, 39444$$The details are as follows (permalink):
- \(12122 = 2 \times 11 \times 19 \times 29 \)
- \(12123 = 3^3 \times 449 \)
- \(12124 = 2^2 \times 7 \times 433 \)
- \(12125 = 5^3 \times 97 \)
- \(12126 = 2 \times 3 \times 43 \times 47 \)
- \(14748 = 2^2 \times 3 \times 1229 \)
- \(14749 = 7^3 \times 43 \)
- \(14750 = 2 \times 5^3 \times 59 \)
- \(14751 = 3^2 \times 11 \times 149 \)
- \(14752 = 2^5 \times 461 \)
- \(17574 = 2 \times 3 \times 29 \times 101 \)
- \(17575 = 5^2 \times 19 \times 37 \)
- \(17576 = 2^3 \times 13^3 \)
- \(17577 = 3^4 \times 7 \times 31 \)
- \(17578 = 2 \times 11 \times 17 \times 47 \)
- \(19940 = 2^2 \times 5 \times 997 \)
- \(19941 = 3 \times 17^2 \times 23 \)
- \(19942 = 2 \times 13^2 \times 59 \)
- \(19943 = 7^2 \times 11 \times 37 \)
- \(19944 = 2^3 \times 3^2 \times 277 \)
- \(22382 = 2 \times 19^2 \times 31 \)
- \(22383 = 3^3 \times 829 \)
- \(22384 = 2^4 \times 1399 \)
- \(22385 = 5 \times 11^2 \times 37 \)
- \(22386 = 2 \times 3 \times 7 \times 13 \times 41 \)
- \(27872 = 2^5 \times 13 \times 67 \)
- \(27873 = 3^2 \times 19 \times 163 \)
- \(27874 = 2 \times 7 \times 11 \times 181 \)
- \(27875 = 5^3 \times 223 \)
- \(27876 = 2^2 \times 3 \times 23 \times 101 \)
- \(28592 = 2^4 \times 1787 \)
- \(28593 = 3^4 \times 353 \)
- \(28594 = 2 \times 17 \times 29^2 \)
- \(28595 = 5 \times 7 \times 19 \times 43 \)
- \(28596 = 2^2 \times 3 \times 2383 \)
- \(29886 = 2 \times 3 \times 17 \times 293 \)
- \(29887 = 11^2 \times 13 \times 19 \)
- \(29888 = 2^6 \times 467 \)
- \(29889 = 3^6 \times 41 \)
- \(29890 = 2 \times 5 \times 7^2 \times 61 \)
- \(35072 = 2^8 \times 137 \)
- \(35073 = 3^4 \times 433 \)
- \(35074 = 2 \times 13 \times 19 \times 71 \)
- \(35075 = 5^2 \times 23 \times 61 \)
- \(35076 = 2^2 \times 3 \times 37 \times 79 \)
- \(35124 = 2^2 \times 3 \times 2927 \)
- \(35125 = 5^3 \times 281 \)
- \(35126 = 2 \times 7 \times 13 \times 193 \)
- \(35127 = 3^3 \times 1301 \)
- \(35128 = 2^3 \times 4391 \)
- \(39444 = 2^2 \times 3 \times 19 \times 173 \)
- \(39445 = 5 \times 7^3 \times 23 \)
- \(39446 = 2 \times 11^2 \times 163 \)
- \(39447 = 3^4 \times 487 \)
- \(39448 = 2^3 \times 4931 \)
Up to ten million, the first runs of \( \textbf{six}\) numbers with \( \textbf{four} \) or more factors, counting multiplicity, start with 7451871, 8813580 and 8961325. Here are the details:
- \(7451871 = 3 \times 7^2 \times 163 \times 311 \)
- \(7451872 = 2^5 \times 232871 \)
- \(7451873 = 11 \times 13 \times 31 \times 41^2 \)
- \(7451874 = 2 \times 3^2 \times 37 \times 67 \times 167 \)
- \(7451875 = 5^4 \times 11923 \)
- \(7451876 = 2^2 \times 19 \times 71 \times 1381 \)
- \(8813580 = 2^2 \times 3 \times 5 \times 146893 \)
- \(8813581 = 7^2 \times 43 \times 47 \times 89 \)
- \(8813582 = 2 \times 17 \times 53 \times 67 \times 73 \)
- \(8813583 = 3^3 \times 197 \times 1657 \)
- \(8813584 = 2^4 \times 13 \times 42373 \)
- \(8813585 = 5 \times 11 \times 37 \times 61 \times 71 \)
- \(8961325 = 5^2 \times 31^2 \times 373 \)
- \(8961326 = 2 \times 11 \times 37 \times 101 \times 109 \)
- \(8961327 = 3^3 \times 61 \times 5441 \)
- \(8961328 = 2^4 \times 560083 \)
- \(8961329 = 13 \times 17 \times 23 \times 41 \times 43 \)
- \(8961330 = 2 \times 3 \times 5 \times 7 \times 139 \times 307 \)
If we consider runs of \( \textbf{six}\) numbers with exactly \( \textbf{three}\) not necessarily distinct prime factors, then in the range up to 40000 the numbers 2522, 4921, 18241, 25553 and 27290 begin these runs (permalink).
If we consider runs of \( \textbf{three}\) numbers with exactly \( \textbf{two}\) not necessarily distinct prime factors, then in the range up to 40000 there are 189 such numbers. The numbers beginning these runs are shown below (permalink):
33, 85, 93, 121, 141, 201, 213, 217, 301, 393, 445, 633, 697, 841, 921, 1041, 1137, 1261, 1345, 1401, 1641, 1761, 1837, 1893, 1941, 1981, 2101, 2181, 2217, 2305, 2361, 2433, 2461, 2517, 2641, 2721, 2733, 3097, 3385, 3601, 3693, 3865, 3901, 3957, 4285, 4413, 4533, 4593, 4881, 5601, 5721, 5853, 5997, 6157, 6241, 6457, 7113, 7141, 7165, 7233, 7341, 7401, 7861, 7977, 8157, 8185, 8257, 8401, 8457, 8913, 9121, 9753, 9937, 9985, 10117, 10237, 11013, 11181, 11281, 11301, 11377, 11641, 11721, 11733, 11757, 12021, 12057, 12777, 13645, 13917, 13953, 14037, 14253, 14901, 14917, 14961, 14977, 14997, 15117, 15177, 15837, 16161, 16177, 16293, 16321, 16437, 16593, 17245, 17337, 17461, 17637, 17857, 18021, 18085, 18453, 18805, 18861, 19101, 19561, 19657, 19713, 19741, 19857, 20017, 20157, 20197, 20281, 21477, 22233, 22297, 22521, 22821, 23377, 24501, 24537, 25105, 25293, 26517, 26581, 26797, 27381, 27517, 27561, 27661, 27717, 27841, 28021, 28113, 28801, 28893, 29037, 29065, 29305, 29517, 29701, 29901, 30297, 30397, 30453, 30541, 30901, 30993, 31285, 31461, 31497, 31917, 32133, 32365, 33481, 34197, 34413, 34777, 34861, 35101, 35193, 35781, 35821, 35857, 35941, 36121, 37041, 37437, 37837, 38137, 38161, 38937, 39001, 39361, 39685
There cannot be runs of more than three numbers with two distinct prime factors because every fourth number will be a multiple of \(4 = 2 \times 2\).
