When Sequences Overlap

I've written in previous posts about what I've called the Factors Sequence (with and without multiplicity) and the Divisors Sequence:

• Number's Divisors to Sequence Algorithm
• Number's Factors to Sequence Algorithm 2
• Number's Factors to Sequence Algorithm 1

What I noticed about the number (28268) associated with my diurnal age today is that for the Factors Sequence with multiplicity and the Divisors Sequence both terminate where they began. Here is the Factors Sequence: 28268, 7067, 14134, 42402, 169608, 28268. Figure 1 shows a plot of the points with a logarithmic vertical axis.

Figure 1

Here is Divisors Sequence: 28268, 339216, 13568640, 106005, 1696080, 21201, 169608, 5427456, 390776832, 2035296, 28268. Figure 2 shows a plot of the points with a logarithmic vertical axis:

Figure 2

Closer inspection of both sequences shows that they share a common number apart from 28268. The number is 169608 and Figure 3 shows both plots on the same graph.

Figure 3: permalink

In the range up 40000, there are 1779 numbers with these two properties:

• the factors sequence with multiplicity and the divisors sequence terminate where they started
• the two sequences share a common member different from the start and end values

Here are the numbers (permalink) from 28000 to 40000:

28001, 28012, 28019, 28036, 28056, 28124, 28132, 28148, 28163, 28204, 28244, 28248, 28252, 28268, 28292, 28307, 28324, 28348, 28364, 28372, 28388, 28396, 28444, 28468, 28492, 28499, 28532, 28536, 28564, 28588, 28612, 28628, 28643, 28676, 28684, 28724, 28732, 28776, 28793, 28796, 28804, 28868, 28892, 28901, 28924, 28949, 28961, 28964, 28968, 28979, 28996, 29021, 29033, 29044, 29064, 29084, 29092, 29108, 29116, 29123, 29156, 29164, 29201, 29212, 29243, 29252, 29256, 29276, 29308, 29339, 29356, 29372, 29428, 29444, 29452, 29453, 29468, 29483, 29492, 29516, 29548, 29564, 29588, 29612, 29636, 29684, 29692, 29708, 29756, 29764, 29784, 29788, 29812, 29832, 29852, 29873, 29884, 29928, 29932, 29972, 30004, 30052, 30072, 30076, 30124, 30172, 30203, 30264, 30269, 30284, 30323, 30347, 30388, 30389, 30404, 30408, 30436, 30449, 30452, 30467, 30476, 30504, 30508, 30524, 30532, 30539, 30548, 30552, 30604, 30644, 30652, 30689, 30716, 30773, 30788, 30803, 30836, 30844, 30884, 30916, 30956, 30988, 31004, 31052, 31076, 31084, 31124, 31132, 31139, 31148, 31196, 31204, 31228, 31244, 31252, 31253, 31259, 31276, 31324, 31348, 31396, 31436, 31444, 31469, 31484, 31512, 31547, 31564, 31588, 31612, 31636, 31649, 31652, 31721, 31756, 31793, 31828, 31844, 31868, 31876, 31916, 31924, 31964, 31988, 31992, 31996, 32003, 32009, 32012, 32084, 32088, 32092, 32108, 32132, 32136, 32141, 32188, 32204, 32228, 32232, 32252, 32284, 32308, 32332, 32376, 32381, 32424, 32476, 32507, 32516, 32524, 32548, 32561, 32564, 32568, 32572, 32573, 32596, 32603, 32612, 32633, 32636, 32692, 32712, 32756, 32788, 32789, 32804, 32812, 32828, 32843, 32852, 32908, 32933, 32955, 32987, 32996, 33004, 33028, 33044, 33053, 33068, 33096, 33107, 33116, 33196, 33236, 33284, 33288, 33308, 33332, 33347, 33356, 33364, 33384, 33388, 33404, 33428, 33432, 33436, 33461, 33484, 33521, 33528, 33532, 33569, 33596, 33604, 33628, 33644, 33652, 33668, 33672, 33713, 33749, 33764, 33773, 33809, 33812, 33836, 33864, 33884, 33892, 33916, 33932, 33941, 33956, 33964, 33988, 34008, 34012, 34028, 34036, 34076, 34124, 34196, 34204, 34228, 34244, 34253, 34268, 34316, 34348, 34372, 34412, 34444, 34468, 34484, 34532, 34556, 34584, 34604, 34612, 34628, 34636, 34667, 34732, 34844, 34868, 34913, 34949, 34968, 34972, 34996, 35036, 35068, 35069, 35081, 35092, 35108, 35164, 35188, 35204, 35236, 35252, 35256, 35339, 35363, 35372, 35404, 35428, 35448, 35492, 35516, 35524, 35564, 35573, 35596, 35612, 35636, 35668, 35684, 35708, 35756, 35788, 35812, 35908, 35924, 35932, 35933, 35956, 35963, 35993, 36024, 36068, 36076, 36083, 36092, 36124, 36148, 36168, 36188, 36212, 36284, 36292, 36299, 36308, 36312, 36316, 36332, 36353, 36356, 36388, 36404, 36408, 36428, 36452, 36467, 36476, 36484, 36501, 36524, 36572, 36596, 36629, 36668, 36676, 36683, 36696, 36716, 36761, 36772, 36788, 36821, 36844, 36868, 36888, 36892, 36916, 36923, 36929, 36932, 36984, 36988, 37012, 37013, 37036, 37049, 37052, 37084, 37148, 37156, 37181, 37196, 37204, 37228, 37252, 37253, 37316, 37388, 37412, 37464, 37516, 37532, 37547, 37556, 37628, 37708, 37796, 37804, 37848, 37853, 37876, 37924, 37948, 37972, 38036, 38068, 38092, 38108, 38116, 38136, 38184, 38189, 38201, 38212, 38228, 38252, 38276, 38284, 38308, 38333, 38356, 38372, 38396, 38428, 38444, 38453, 38468, 38472, 38501, 38548, 38564, 38588, 38603, 38636, 38668, 38669, 38684, 38692, 38732, 38747, 38764, 38804, 38812, 38828, 38852, 38861, 38867, 38873, 38908, 38924, 38933, 39028, 39044, 39052, 39089, 39092, 39107, 39144, 39172, 39188, 39192, 39196, 39233, 39236, 39308, 39323, 39336, 39364, 39388, 39412, 39419, 39432, 39452, 39476, 39521, 39524, 39569, 39572, 39576, 39596, 39624, 39652, 39668, 39748, 39772, 39779, 39812, 39827, 39836, 39844, 39864, 39908, 39916, 39932, 39953, 39956, 39964, 39988, 39989

28224: An Interesting Number

28224 has 194 entries in the OEIS which is extraordinarily high for a five digit number. In this post I'll be discussing some of this number's most interesting properties but not all of them. There are just too many. It's prime factorisation is:$$28224=2^6 \times 3^2 \times 7^2$$FIRST INTERESTING PROPERTY

Numbers that are perfect squares are quite rare in the range up to 40000. There are only 200 of them and 28224, my diurnal age today, is one of them. It has the property that:$$28224=168^2$$The number of days between my experience of them is a little less than a year. There is a gap of exactly 365 days between \(183^2\) and \(182^2\) since:$$ \begin{align} 183^2-182^2 &= (183 + 182)(183-182) \\ &=365 \times 1 \\ &=365 \end{align}$$I'll be \(33124\) or \(182^2\) days when I'm over \(90\) years old so I may not get to experience the transition from this square to the next.

28224 is also a Loeschian number since it is equal to \(72^2+ 72 \times 120 + 120^2\).

28224 also has a product of digits (256) that is a perfect square since \(256=16^2\).

SECOND INTERESTING PROPERTY

Numbers that are the sum of two positive cubes are relatively rare in the range up to 40000. In fact, there are only 378 numbers in the range up to 40000 and 28224 is one of them because:$$28224=22^3 + 26^3$$These numbers form OEIS A004999.

THIRD INTERESTING NUMBER

Energetic numbers are numbers that can be broken into two or more substrings and expressed as a sum of (possibly different) positive powers of those substrings. They form OEIS  A055480. 28224 is one such number because:$$28224=28^3 + 2^{11} + 2^7 + 4^6$$I discuss this category of numbers in my blog post Energetic Numbers.

FOURTH INTERESTING NUMBER

Friedman numbers are positive integers which can be written in some non-trivial way using its own digits, together with the symbols + – × / ^ ( ) and concatenation. 28224 is one such number because:$$28224 = (2 + 82)^2 × 4$$It is said to be a "nice" Friedman number because the digits are in the same order as the number. These numbers are listed on my blog post Narcissistic, D-Powerfull and Friedman Numbers.

FIFTH INTERESTING NUMBER

28224 has the property that certain of its factors (not necessarily prime) can be arranged to form a palindrome. Specifically:$$2 \times 2 \times 2 \times 882 \times 2 \times 2 = 22288222$$I've written about these sorts of numbers in a post titled Why Is 313131 An Interesting Number?

SIXTH INTERESTING PROPERTY

28224 is a concatenation of powers of 2 since:$$28224= 2^1 \; || \; 2^3 \; || \; 2^1 \; || \; 2^1 \; || \; 2^2$$I've written about numbers that can be formed in this way in a blog post titled Nothing New Under The Sun. It is also a concatenation of multiples of 7 since:$$28224= (7 \times 4) \, || \, (7 \times 32)$$I posted about these sorts of concatenations in my blog post More Numbers as Concatenations.

SEVENTH INTERESTING PROPERTY

28224 is a member of OEIS A253824 where$$ \text{numbers } m = s \, || \, t \text{ such that } m = \sigma(s) \times \sigma(t)$$where || represents concatenation. In the case of 28224 we have:$$ \begin{align} 28224 &= 28 \, || \, 224 \\ &= \sigma(28) \times \sigma(224) \\ &= 56 \times 504 \\ &=28224 \end{align}$$28224 is only the third such number in the range up to 40000. The two earlier numbers are 540 and 2352.

EIGHTH INTERESTING PROPERTY

28224 has a digit sum of 18 and when this is added to the number the result is 28242 which has the same digits as 28224 but in a slightly different order. This property makes it a member of OEIS A246420.

A246420

Numbers \(n\) such that \(n\)  + digit sum of \(n\) is a permutation of the decimal digits of \(n\) .

Forming Palindromes from Factors

 In a blog post titled Why Is 313131 An Interesting Number?, I remarked that:

$$ 313131=3 \times 7 \times 13 \times 31 \times 37$$If we rearrange the order of multiplication we get the following:$$ 313131=7 \times 3 \times 13 \times 31 \times 37$$Concatenating these digits we get the number \(73133137\) which is palindromic.

I went on to look at what other numbers in the range between 28000 and 29000 have this property and came up with the table shown below:

The numbers are thus relatively rare, there being only 27 in a range of 1000 numbers. This represents a density of 2.7%. The numbers are listed below:

28072, 28125, 28194, 28224, 28242, 28273, 28308, 28322, 28332, 28416, 28431, 28448, 28585, 28589, 28593, 28601, 28602, 28609, 28620, 28672, 28685, 28692, 28750, 28800, 28812, 28847, 28951

The factors under consideration here are all PRIME factors. What if we allow factors that are not necessarily prime. Take 28200 as an example:$$ \begin{align} 28200 &= 2 \times 5 \times 3 \times 2 \times 235 \times 2 \\ &\rightarrow 25322352 \end{align}$$The resultant number after concatenation of the factors is palindromic. Notice that the factor 235 is NOT prime.

It turns out that palindromes constructed in this way are relatively frequent. In the range between 28100 and 28300 (a range of only 200), the density is 15.4%. The numbers are:

28104, 28105, 28125, 28126, 28128, 28130, 28140, 28143, 28152, 28160, 28161, 28175, 28179, 28180, 28182, 28188, 28194, 28200, 28224, 28230, 28236, 28242, 28251, 28256, 28266, 28273, 28275, 28280, 28288, 28296, 28300

I've set up my multipurpose algorithm to identify such numbers when they pop up in my diurnal age analysis.

Pandigital Products

Yesterday I turned 28156 days old and this number has an interesting property:$$28156 = 4 \times 7039$$The factorisation shown is not the prime factorisation but, looking at both sides of the equation, it can be seen that each of the digits from 0 to 9 occurs exactly once. This makes the number a member of OEIS A370970:

A370970: numbers \(k\) which have a factorization \(k = f_1 \times f_2 \times \ldots \times f_n \) where the digits of \({k, f_1, f_2, \ldots, f_n}\) together give \(0,1, \ldots ,9\) exactly once.

Here is the complete list of terms:

8596 = 2 x 14 x 307

8790 = 2 x 3 x 1465

9360 = 2 x 4 x 15 x 78

9380 = 2 x 5 x 14 x 67

9870 = 2 x 3 x 1645

10752 = 3 x 4 x 896

12780 = 4 x 5 x 639

14760 = 5 x 9 x 328

14820 = 5 x 39 x 76

15628 = 4 x 3907

15678 = 39 x 402

16038 = 27 x 594 = 54 x 297

16704 = 9 x 32 x 58

17082 = 3 x 5694

17820 = 36 x 495 = 45 x 396

17920 = 8 x 35 x 64

18720 = 4 x 5 x 936

19084 = 52 x 367

19240 = 8 x 37 x 65

20457 = 3 x 6819

20574 = 6 x 9 x 381

20754 = 3 x 6918

21658 = 7 x 3094

24056 = 8 x 31 x 97

24507 = 3 x 8169

25803 = 9 x 47 x 61

26180 = 4 x 7 x 935

26910 = 78 x 345

27504 = 3 x 9168

28156 = 4 x 7039

28651 = 7 x 4093

30296 = 7 x 8 x 541

30576 = 8 x 42 x 91

30752 = 4 x 8 x 961

31920 = 5 x 76 x 84

32760 = 8 x 45 x 91

32890 = 46 x 715

34902 = 6 x 5817

36508 = 4 x 9127

47320 = 8 x 65 x 91

58401 = 63 x 927

65128 = 7 x 9304 

65821 = 7 x 9403

These numbers are few and far between as can be seen and 28156 in particular recurs with permuted digits as 15628, 21658, 28651, 65128 and 65821.

Concatenating Reversible Sphenic Numbers

My previous post Concatenating Emirpimes suggested the concatenation of reversible sphenic numbers as a logical sequel. A sphenic number that is still sphenic when its digits are reversed is sometimes called a \( \textbf{cinephs} \). Let's take 418 as an example of the type of number that we want to identify:$$ \begin{align} 418 &= 2 \times 11 \times 19 \\ 814 &= 2 \times 11 \times 37 \\ 418814 &= 2 \times 11 \times 19037 \end{align}$$There are, coincidentally, 418 such numbers in the range up to 40000. Here are the details (permalink):

=== Cineph Concatenation Statistics ===

Range evaluated: 1 to 40000

1. Sphenic numbers: 7720 (19.30% of the range)

2. Cinephs: 1890 (4.72% of the range)

3. Successful Cinephs: 418 (1.0450% of the range)

=== Comma-Separated List of Successful Cinephs ===

418, 682, 759, 814, 957, 1034, 1095, 1113, 1185, 1342, 1419, 1446, 1606, 1614, 1902, 1965, 2014, 2035, 2282, 2414, 2431, 2438, 2494, 2635, 2665, 2686, 3018, 3059, 3138, 3201, 3278, 3297, 3358, 3495, 3597, 3606, 3678, 3685, 3714, 3729, 3765, 3926, 4301, 4382, 4543, 4565, 4715, 4945, 5174, 5362, 5495, 5863, 5986, 6035, 6061, 6083, 6206, 6293, 6806, 7185, 7305, 7414, 7449, 7567, 7645, 7657, 7662, 7718, 8155, 8174, 8378, 8386, 8390, 8393, 8555, 8734, 8786, 8789, 8987, 9138, 9141, 9213, 9309, 9321, 9354, 9362, 9503, 9515, 9519, 9807, 9822, 9885, 9951, 9978, 10095, 10246, 10263, 10274, 10326, 10382, 10490, 10502, 10509, 10554, 10635, 10761, 10835, 10879, 10915, 11098, 11342, 11395, 11398, 11407, 11577, 11605, 11753, 11922, 11937, 11958, 11982, 12174, 12218, 12257, 12378, 12498, 12529, 12551, 12595, 12859, 12874, 12890, 12914, 13137, 13143, 13222, 13305, 13514, 13574, 13618, 13634, 13783, 13786, 13906, 14035, 14043, 14055, 14127, 14174, 14223, 14298, 14313, 14443, 14573, 14590, 14619, 14646, 14655, 14846, 14889, 14894, 14955, 15235, 15323, 15422, 15430, 15521, 15585, 15626, 15657, 15794, 15818, 15914, 15958, 15994, 16098, 16159, 16518, 16558, 16666, 16701, 16914, 16915, 16946, 17006, 17139, 17353, 17445, 17518, 17534, 17799, 17805, 17814, 17818, 17978, 18093, 18123, 18147, 18205, 18222, 18231, 18309, 18395, 18458, 18555, 18579, 18685, 18717, 18814, 18854, 19023, 19178, 19221, 19317, 19382, 19545, 19605, 19623, 19877, 19923, 19947, 19987, 20009, 20066, 20195, 20270, 20305, 20315, 20530, 20594, 20657, 20726, 20758, 20870, 20905, 20945, 20965, 20966, 22058, 22282, 22562, 22582, 22591, 22645, 22933, 22939, 22945, 22970, 24002, 24026, 24095, 24265, 24409, 24479, 24583, 24590, 24739, 24817, 24878, 24962, 24973, 26090, 26102, 26134, 26146, 26245, 26266, 26273, 26429, 26494, 26506, 26546, 26555, 26614, 26638, 26663, 26758, 26873, 26878, 26930, 26939, 26990, 28085, 28165, 28186, 28222, 28237, 28358, 28418, 28426, 28441, 28483, 28514, 28645, 28747, 28835, 28870, 28918, 30054, 30099, 30126, 30277, 30418, 30502, 30651, 30795, 30943, 31026, 31027, 31035, 31191, 31254, 31386, 31658, 31719, 31978, 32039, 32043, 32181, 32295, 32351, 32417, 32442, 32613, 32846, 32857, 32898, 32907, 32938, 33143, 33162, 33186, 33378, 33429, 33473, 33555, 33583, 33585, 33754, 33818, 33835, 33906, 33918, 33922, 34005, 34077, 34078, 34131, 34133, 34158, 34203, 34222, 34257, 34474, 34562, 34563, 34661, 34705, 34826, 35006, 35094, 35123, 35202, 35274, 35319, 35382, 35618, 35619, 35871, 35949, 35985, 36003, 36039, 36165, 36174, 36226, 36355, 36474, 36586, 36606, 36627, 36935, 36958, 37317, 37329, 37367, 37411, 37418, 37433, 37527, 37614, 37634, 37743, 37754, 37970, 38049, 38085, 38193, 38294, 38399, 38451, 38467, 38533, 38555, 38643, 39081, 39169, 39305, 39306, 39351, 39358, 39399, 39458, 39562, 39785, 39842, 39918, 39966

As we learned in my previous post: 

There is a fundamental rule in number theory: Every palindrome with an even number of digits is divisible by 11.

Thus all the above numbers have 11 as a prime factor. Let's take 39358 as another example: $$ \begin{align} 39358 &= 2 \times 11 \times 1789 \\ 85393 &= 7 \times 11 \times 1109 \\ 3935885393 &= 11 \times 397 \times 901279\end{align}$$Of course, if we were to try this technique on emirps (reversible primes), we would never end up with a prime because the resultant number would always be divisible by 11. For example, 37 and 73 combine to form$$ 3773 = 11 \times 343 =11 \times 7^3$$

Fibonacci From Prime Factors

Gemini's Infographic Summary of the Content in this Post

I noticed that the number (\( \textbf{28075} \)) associated with my diurnal age today has an interesting property relating to its prime factors:$$28075=5^2 \times 1123$$The number has distinct prime factors of \( \textbf{5}\) and \( \textbf{1123}\). If these two factors are written in reversed order and then concatenated, the number \( \textbf{11235}\) is formed with digits that form a Fibonacci sequence:$$1+1 \rightarrow 2 \text{ and } 2 + 3 \rightarrow 5$$This got me thinking about what other numbers have this property and so I set Gemini to work to find all such numbers in the range from 1 to 40000. It turns out that the following numbers qualify (permalink):

22, 26, 30, 44, 52, 60, 66, 70, 88, 90, 101, 104, 115, 120, 132, 140, 141, 150, 158, 167, 176, 180, 198, 203, 205, 208, 210, 240, 242, 253, 257, 264, 270, 280, 300, 301, 316, 330, 338, 347, 350, 352, 360, 396, 416, 420, 423, 427, 450, 480, 484, 490, 528, 540, 560, 575, 594, 600, 611, 617, 630, 632, 660, 676, 700, 704, 720, 726, 750, 771, 790, 792, 810, 832, 835, 840, 900, 960, 968, 980, 990, 1025, 1050, 1056, 1080, 1120, 1123, 1188, 1200, 1222, 1260, 1264, 1265, 1269, 1320, 1350, 1352, 1400, 1408, 1421, 1440, 1452, 1459, 1470, 1500, 1580, 1584, 1620, 1650, 1664, 1680, 1750, 1782, 1800, 1890, 1920, 1936, 1960, 1980, 2100, 2107, 2112, 2160, 2178, 2240, 2250, 2313, 2376, 2400, 2430, 2444, 2450, 2520, 2528, 2640, 2645, 2662, 2700, 2704, 2783, 2800, 2816, 2875, 2880, 2904, 2940, 2970, 2989, 3000, 3150, 3160, 3168, 3240, 3257, 3300, 3328, 3360, 3430, 3500, 3564, 3600, 3630, 3750, 3780, 3807, 3840, 3872, 3920, 3950, 3960, 4050, 4175, 4200, 4224, 4320, 4356, 4377, 4394, 4410, 4480, 4500, 4752, 4800, 4860, 4888, 4900, 4950, 5040, 5056, 5125, 5167, 5250, 5279, 5280, 5324, 5346, 5400, 5408, 5600, 5615, 5632, 5670, 5760, 5808, 5819, 5880, 5887, 5940, 6000, 6300, 6319, 6320, 6325, 6336, 6480, 6534, 6600, 6627, 6656, 6720, 6750, 6860, 6939, 7000, 7128, 7200, 7260, 7290, 7350, 7500, 7560, 7680, 7744, 7840, 7900, 7920, 7943, 7986, 8100, 8250, 8400, 8405, 8448, 8640, 8712, 8750, 8788, 8820, 8910, 8960, 9000, 9450, 9504, 9600, 9720, 9776, 9800, 9900, 9947, 10080, 10112, 10201, 10290, 10500, 10560, 10648, 10692, 10800, 10816, 10890, 11200, 11250, 11264, 11340, 11421, 11520, 11616, 11760, 11880, 12000, 12150, 12250, 12482, 12600, 12640, 12672, 12943, 12960, 13068, 13131, 13200, 13225, 13230, 13312, 13440, 13500, 13720, 13915, 14000, 14256, 14375, 14400, 14520, 14580, 14700, 14749, 14850, 15000, 15120, 15360, 15488, 15680, 15750, 15800, 15840, 15886, 15972, 16038, 16200, 16500, 16800, 16896, 17010, 17150, 17280, 17424, 17500, 17576, 17640, 17820, 17920, 18000, 18150, 18750, 18900, 19008, 19200, 19440, 19552, 19600, 19602, 19750, 19800, 19881, 20160, 20224, 20250, 20580, 20817, 20875, 20923, 21000, 21120, 21296, 21347, 21384, 21600, 21632, 21780, 21870, 22050, 22400, 22500, 22528, 22680, 23040, 23232, 23520, 23760, 23958, 24000, 24010, 24300, 24500, 24750, 24964, 25200, 25280, 25344, 25625, 25920, 26047, 26136, 26250, 26400, 26460, 26624, 26730, 26880, 27000, 27440, 27889, 28000, 28075, 28350, 28512, 28717, 28800, 29040, 29095, 29160, 29282, 29400, 29700, 30000, 30240, 30613, 30720, 30870, 30976, 31360, 31500, 31600, 31625, 31680, 31772, 31944, 32076, 32400, 32670, 33000, 33600, 33750, 33792, 34020, 34263, 34300, 34560, 34848, 35000, 35152, 35280, 35640, 35840, 36000, 36300, 36450, 36750, 37500, 37800, 38016, 38400, 38880, 39104, 39200, 39204, 39393, 39500, 39600, 39690, 39930

Of course looking at these numbers it's not immediately apparent what the Fibonacci digit sequence is but the Gemini program creates a table to show this. I'll restrict the range to between 28000 and 29000. The result is shown in Figure 1:

Figure 1: permalink

The fact that we are only considering \( \textbf{distinct} \) prime factors helps the program run quickly and there are no problems using it with SageMathCell. However if we allow multiplicity of factors, the number of permutations increases dramatically and SageMathCell quickly times out even if we restrict the range to between 28000 and 29000. So I think working only with distinct prime factors is the way to go.

The Revision Revisited

In my post titled A Revision, I outlined an alternative algorithm for building a sequence based on the factors of numbers considered with multiplicity. This is the algorithm:

Suppose we take any positive integer \(n \gt 1\) and apply the following rules to it:

• if a \(4k+1\) prime, double it and add 1: \(n \rightarrow 2n+1\)
• if a \(4k+3\) prime, subtract 1 and divide by 2: \(n \rightarrow (n-1)/2\)
• if composite, determine its number of factors \(f\) counted \( \textbf{with multiplicity}\)
• if \( n \pmod f \equiv 0\) then \(n \rightarrow \dfrac{n}{f} \)
• if \( n \pmod f \not\equiv 0 \) then \(n \rightarrow n \times f\)

Keep repeating this process until a loop is reached or call a stop after a fixed number of iterations. 

What I'm interested in looking at in this post are the record lengths of sequences generated over a range of numbers, noting as well the highest and lowest values that are reached by sequence members. Figure 1 shows the results in the range from one to one million (permalink):

Figure 1

Let's now consider the algorithm where multiplicity is ignored. This algorithm in as follows:

Suppose we take any positive integer \(n \gt 1\) and apply the following rules to it:

• if a \(4k+1\) prime, double it and add 1: \(n \rightarrow 2n+1\)
• if a \(4k+3\) prime, subtract 1 and divide by 2: \(n \rightarrow (n-1)/2\)
• if composite, determine its number of factors \(f\) counted \( \textbf{without multiplicity}\)
• if \( n \pmod f \equiv 0\) then \(n \rightarrow \dfrac{n}{f} \)
• if \( n \pmod f \not\equiv 0 \) then \(n \rightarrow n \times f\)

Keep repeating this process until a loop is reached or call a stop after a fixed number of iterations. 

Figure 2 shows the record lengths, highs and lows for the range from one to one million (permalink).

Figure 2

It can be noted that the record lengths of sequences are shorter when multiplicity is ignored and more end in 1 rather than a loop.

A Revision

In my previous post, I made a modification to the following algorithm:

Suppose we take any positive integer \(n \gt 1\) 

• if prime, double it and add 1: \(n \rightarrow 2n+1\)
• if composite, determine its number of divisors \(d\)
• if \( n \pmod d \equiv 0\) then \(n \rightarrow \dfrac{n}{d} \)
• if \( n \pmod d \not\equiv 0 \) then \(n \rightarrow n \times d\)

Keep repeating this process until a loop is reached or call a stop after a fixed number of iterations. 

The modification I made prevented the numbers generated from becoming too large too quickly. Instead of doubling a prime and adding 1, I decided to do this only if the number was a \(4k+1\) prime. If it was a \(4k+3\), I subtracted 1 and divided by 2. The new algorithm looks like this:

• if a \(4k+1\) prime, double it and add 1: \(n \rightarrow 2n+1\)
• if a \(4k+3\) prime, subtract 1 and divide by 2: \(n \rightarrow (n-1)/2\)
• if composite, determine its number of divisors \(d\)
• if \( n \pmod d \equiv 0\) then \(n \rightarrow \dfrac{n}{d} \)
• if \( n \pmod d \not\equiv 0 \) then \(n \rightarrow n \times d\) 

Here is an example using 28069 (permalink):

--- Loop detected at value 149708 ---

Divisors to Sequence:

28069, 56139, 224556, 18713, 37427, 149708, 1796496, 71859840, 561405, 8982480, 112281, 898248, 28743936, 2069563392, 10778976, 149708

------------------------------

Sequence Length: 16

Highest Value:   2069563392

This method of dealing with primes is also better suited to the sequences I mentioned earlier in my two posts: 

• Number's Factors to Sequence Algorithm 1
• Number's Factors to Sequence Algorithm 2

Here are the two new algorithms. 

Suppose we take any positive integer \(n \gt 1\) and apply the following rules to it:

• if a \(4k+1\) prime, double it and add 1: \(n \rightarrow 2n+1\)
• if a \(4k+3\) prime, subtract 1 and divide by 2: \(n \rightarrow (n-1)/2\)
• if composite, determine its number of factors \(f\) counted \( \textbf{with multiplicity}\)
• if \( n \pmod f \equiv 0\) then \(n \rightarrow \dfrac{n}{f} \)
• if \( n \pmod f \not\equiv 0 \) then \(n \rightarrow n \times f\)

Keep repeating this process until a loop is reached or call a stop after a fixed number of iterations. 

Here is an example using 28069 (permalink):

--- Loop detected at value 37427 ---

Number of factors to sequence with multiplicity:

28069, 56139, 112278, 37426, 18713, 37427, 74854, 224562, 898248, 149708, 37427

--------------------

Sequence Length: 11 

Highest Value:   898248 

Suppose we take any positive integer \(n \gt 1\) and apply the following rules to it:

• if a \(4k+1\) prime, double it and add 1: \(n \rightarrow 2n+1\)
• if a \(4k+3\) prime, subtract 1 and divide by 2: \(n \rightarrow (n-1)/2\)
• if composite, determine its number of factors \(f\) counted \( \textbf{without multiplicity}\)
• if \( n \pmod f \equiv 0\) then \(n \rightarrow \dfrac{n}{f} \)
• if \( n \pmod f \not\equiv 0 \) then \(n \rightarrow n \times f\)

Keep repeating this process until a loop is reached or call a stop after a fixed number of iterations. 

Here is an example using 28069 (permalink):

--- Loop detected at value 224562 ---

28069, 56139, 112278, 37426, 18713, 37427, 74854, 224562, 898248, 224562

--------------------

Sequence Length: 10

Highest Value:   898248

Number's Factors to Sequence Algorithm 2

 Suppose we take any positive integer \(n \gt 1\) and apply the following rules to it:

• if prime, double it and add 1: \(n \rightarrow 2n+1\)
  
• if composite, determine its number of factors \(f\) counted \( \textbf{without} \) multiplicity
  
• if \( n \pmod f \equiv 0\) then \(n \rightarrow \dfrac{n}{f} \)
  
• if \( n \pmod f \not\equiv 0 \) then \(n \rightarrow n \times f\)

Keep repeating this process until a loop is reached or call a stop after a fixed number of iterations. This process is exactly the same as in my previous post except that the number of factors is counted without multiplicity. Let's apply this algorithm to 28059. The result is the sequence 28059, 9353, 18706, 56118, 224472, 56118. Here are the details (permalink):

• \(28059 = 3 \times 47 \times 199\) with three factors
  3 divides 28059 to give 9353
  
  
• \(9353 = 47 \times 199\) with two factors but 2 doesn't divide 9353
  multiplying by 2 gives 18706
  
  
• \(18706 = 2 \times 47 \times 199\) with three factors but 3 doesn't divide 18706
  multiplying by 3 gives 56118
  
  
• \(56118 = 2 \times 3 \times 47 \times 199\) with four factors but 4 doesn't divide 56118
  multiplying by 4 gives 224472
  
  
• \(224472 = 2^3 \times 3 \times 47 \times 199\) with four distinct prime factors
  4 divided into 224472 gives 56118
  
  
• \(56118\) occurred earlier in the sequence and so we have a loop

The fluctuations between terms are less extreme and the record breaking numbers this time around are 2, 3, 6, 12, 24, 31, 62, 93, 139, 278, 417, 1251, 3753, 8896, 17792, 18433, 36866, 55299, 165897, 248851, 497702, 746553.

Here are the sequence lengths of these record breakers (permalink) up to one million:

2 --> 10

3 --> 14

6 --> 15

12 --> 16

24 --> 17

31 --> 18

62 --> 19

93 --> 21

139 --> 23

278 --> 24

417 --> 26

1251 --> 27

3753 --> 28

8896 --> 29

17792 --> 30

18433 --> 31

36866 --> 32

55299 --> 34

165897 --> 35

248851 --> 37

497702 --> 38

746553 --> 40

When the algorithm ran with multiplicity of factors being counted, the maximum sequence length up to one million was 155. The trajectory of the last number in the previous list (746553) is as follows:

746553, 1493106, 497702, 248851, 497703, 995406, 331802, 165901, 331803, 663606, 221202, 73734, 24578, 12289, 24579, 49158, 16386, 5462, 2731, 5463, 10926, 3642, 1214, 607, 1215, 2430, 810, 270, 90, 30, 10, 5, 11, 23, 47, 95, 190, 570, 2280, 570

Figure 1 shows a graph of the trajectory of 746553 using a logarithmic \(y\) scale.

Figure 1

Number's Factors to Sequence Algorithm 1

Suppose we take any positive integer \(n \gt 1\) and apply the following rules to it:

• if prime, double it and add 1: \(n \rightarrow 2n+1\)
  
• if composite, determine its number of factors \(f\) counted with multiplicity
  
• if \( n \pmod f \equiv 0\) then \(n \rightarrow \dfrac{n}{f} \)
• if \( n \pmod f \not\equiv 0 \) then \(n \rightarrow n \times f\)

Keep repeating this process until a loop is reached or call a stop after a fixed number of iterations. Let's use 28058 as an example. The sequence generated is 28058, 14029, 28059, 9353, 18706, 56118, 224472, 37412, 9353 and the details are as follows:

• \(28058 = 2 \times 14029\) and there are two factors
  2 divides 28056 to give 14209
  
  
• \(14029\) is prime
  multiplying by 2 and adding 1 we get 28059
  
  
• \(28059 = 3 \times 47 \times 199\) and there are three factors
  3 divides 28059 to give 9353
  
  
• \(9353 = 47 \times 199\) and there are two factors but 2 doesn't divide 9353
  multiplying by 2 gives 18706
  
  
• \(18706 = 2 \times 47 \times 199\) and there are three factors but 3 doesn't divide 18706
  multiplying by 3 gives 56118
  
  
• \(56118 = 2 \times 3 \times 47 \times 199\) and there are four factors but 4 doesn't divide 56118
  multiplying by 4 gives 224472
  
  
• \(224472 = 2^3 \times 3 \times 47 \times 199\) and there are six factors (with multiplicity)
  6 divides 224472 to give 37412
  
  
• \(37412 = 2^2 \times 47 \times 199\) and there are four factors (with multiplicity)
  4 divides 37412 to give 9353
  
  
• \(9353\) occurred earlier in the sequence and so we have a loop

Figure 1 shows the trajectory.

Figure 1

Some numbers return to their starting points. 27056 is one such number. It's sequence is 28056, 4676, 1169, 2338, 7014, 28056. What appeals to me about this sequence is that it is \( \textbf{base independent}\). Here is permalink to generate the sequence of any number entered into it.

An investigation into what numbers produced sequences of record lengths returned the following number in the range up to one million:

2, 3, 6, 8, 13, 19, 38, 57, 76, 304, 1024, 1579, 2401, 3584, 10331, 12119, 12500, 15379, 24251, 30689, 48661, 57122, 66749, 116603, 155201, 232801, 465602, 698403, 931204

The final number in the list (931204) produces a sequence of length 155. Here are the full details for all the numbers in the list (permalink):

2 --> 11

3 --> 15

6 --> 16

8 --> 18

13 --> 20

19 --> 24

38 --> 25

57 --> 27

76 --> 29

304 --> 32

1024 --> 34

1579 --> 40

2401 --> 43

3584 --> 49

10331 --> 51

12119 --> 53

12500 --> 61

15379 --> 64

24251 --> 65

30689 --> 66

48661 --> 69

57122 --> 92

66749 --> 105

116603 --> 145

155201 --> 146

232801 --> 150

465602 --> 151

698403 --> 153

931204 --> 155

The sequence for 931204 is as follows: 

931204, 2793612, 698403, 1396806, 465602, 232801, 465603, 931206, 310402, 155201, 310403, 1241612, 7449672, 931209, 4656045, 27936270, 223490160, 2458391760, 204865980, 1843793820, 167617620, 16761762, 134094096, 1475035056, 122919588, 13657732, 95604124, 764832992, 69530272, 695302720, 8343632640, 556242176, 7231148288, 516510592, 6198127104, 92971906560, 5468935680, 341808480, 28484040, 256356360, 2819919960, 234993330, 26110370, 182772590, 1462180720, 132925520, 13292552, 1661569, 8307845, 49847070, 398776560, 4386542160, 365545180, 3289906620, 299082420, 29908242, 239265936, 2631925296, 219327108, 1973943972, 179449452, 1794494520, 149541210, 16615690, 2373670, 14242020, 113936160, 1253297760, 104441480, 939973320, 85452120, 8545212, 68361696, 751978656, 62664888, 563983992, 51271272, 512712720, 42726060, 4747340, 33231380, 265851040, 2924361440, 35092337280, 2339489152, 179960704, 2159528448, 32392926720, 550679754240, 30593319680, 458899795200, 26994105600, 1687131600, 140594300, 1265348700, 115031700, 11503170, 92025360, 1012278960, 84356580, 759209220, 69019020, 6901902, 55215216, 607367376, 50613948, 5623772, 803396, 4820376, 602547, 3012735, 18076410, 144611280, 13146480, 1314648, 164331, 821655, 4929930, 39439440, 433833840, 36152820, 4016980, 28118860, 224950880, 20450080, 2045008, 255626, 1278130, 7668780, 61350240, 674852640, 56237720, 506139480, 46012680, 4601268, 36810144, 404911584, 33742632, 303683688, 27607608, 276076080, 23006340, 2556260, 365180, 2191080, 273885, 54777, 219108, 36518, 146072, 876432, 109554, 547770, 91295, 365180

The range of values in this sequence is extreme, ranging from a minimum of 36,518 to a maximum of 550,679,754,240. Figure 2 shows the trajectory with a log scale being necessary for the \(y\) axis.

Figure 2