A Special 29x + 1 Map

 In blog posts titled:

• The 17x+1 Map on the 18th March 2018
• The 13x+1 and 11x+ 1 Maps on the 19th of March of 2018
• The 17x+1 Map Revisited on the 19th May 2025

I've looked at Collatz-style sequences of the form \(px+1\) that map a number \(n\) to the following:

• \(n \rightarrow \dfrac{n}{2}\) if \(2 \, | \, n\)
• \(n \rightarrow \dfrac{n}{3}\) if \(3 \, | \, n\)
• \(n \rightarrow \dfrac{n}{5}\) if \(5 \, | \, n\)
   \(\dots\)
• \(n \rightarrow \dfrac{n}{p-1}\) if \((p-1) \, | \, n\)
• \(n \rightarrow p \times n+1\) \( \text{ if none of the primes less than }p \text{ divide } n\)

Other mappings are of course possible and the following is perhaps the simplest of all. For a given prime \(p\) when applied to a number \(n\), the rule is:

• \(n \rightarrow \dfrac{n}{2}\) if \(n\) is even
• \(n \rightarrow p \times n + 1\) if \(n\) is odd

For \(p=3\), this is the Collatz trajectory of the number. Let's apply this sort of modified \(px+1\) mapping in the case of \(n=3\). The sequence formed by the trajectory up to one million is then (permalink):

3, 88, 44, 22, 11, 320, 160, 80, 40, 20, 10, 5, 146, 73, 2118, 1059, 30712, 15356, 7678, 3839, 111332, 55666, 27833, 807158, 403579, 11703792, 5851896, 2925948, 1462974, 731487 (OEIS A037112)

The trajectory is shown in Figure 1 using a logarithmic scale for the \(y\) axis:

Figure 1: permalink

By contrast if the first mentioned rule above were applied then the trajectory would be short-lived indeed:$$3 \rightarrow  1 \rightarrow  30 \rightarrow  15 \rightarrow 5 \rightarrow  1$$It can be seen that the trajectory very quickly enters a loop. 

Letting \(p=101\) and applying the mapping to \(n=3\) once again, we get the following sequence up to one million formed by the trajectory (permalink):

3, 304, 152, 76, 38, 19, 1920, 960, 480, 240, 120, 60, 30, 15, 1516, 758, 379, 38280, 19140, 9570, 4785, 483286, 241643

The trajectory is shown in Figure 2 using a logarithmic scale for the \(y\) axis:

Figure 2: permalink

Complex Chicanery

I came across an interesting video on YouTube, uploaded today on a channel named Higher Mathematics that posed the following problem:$$ \text{Solve for } x:\\1^x=5$$Clearly the problem has no solution amongst the real numbers but there turns out to be an infinity of solutions once we introduce complex numbers. The solution (with \(k=1,2,3 \dots \) ) then unfolds:$$\begin{align} e^{i \theta} &= \cos{\theta}+i \sin{\theta}\\e^{i 2k \pi} &= \cos{2k \pi}+i \sin{2k \pi}  \\&=1 \\ \therefore (e^{i 2k \pi)})^x &= 5\\ e^{i 2k \pi x} &= 5 \\\ln {e^{i 2k \pi x} }&= \ln{5} \\i 2k \pi x &= \ln{5} \\x &= \frac{ \ln{5}}{i 2k \pi} \\\ &=\frac{-i \ln{5}}{2k \pi}\end{align} $$The complex numbers for values of x given by \(k=1, 2,3\) are shown in Figure 1. They all map to the point \(5+0i\).

Figure 1

Thus we have a function \( f(x) \) with its domain D being the countably infinite set of points:$$ \left( \Big( 0,\dfrac{-i \ln{5}} {2k \pi} \Big) \text{ where k=1, 2, 3, ...} \right) $$ that maps all these points to \( (5, 0i) \) so that:$$ f(x) \rightarrow (5, 0i) \text{ for all x in D}$$

*********************

Let's look now at a slightly different problem:$$2^x=x$$Right from the start, it can be said once again that there are no real solutions to this equality. This is clear once we graph the line \(y=2^x\) and \(y=x\) where we see that there are no points of intersection. See Figure 2.

Figure 2

This problem was posed on the same YouTube channel mentioned earlier. Here is a link to the video. So how do we find if any complex numbers satisfy this equality? Firstly, we take the natural logarithms of both sides and proceed from there:$$ \begin{align} 2^x &= x \\ \ln{2^x} &= \ln{x} \\ x \ln{2} &= \ln{x} \\ \frac{\ln{x}}{x} &= \ln{2} \\ \frac{\ln{x}}{e^{\, \ln{x}} }&= \ln{2} \\ \ln{x} \, {e^{\, -\ln{x}} } &= \ln{2} \\ -\ln{x} \, {e^{\, -\ln{x}} } &= -\ln{2} \\ W(-\ln{x} \, {e^{\, -\ln{x}} } ) &= W(-\ln{2}) \\ -\ln{x} &= W(-\ln{2} ) \\ \ln{x} &= -W(-\ln{2}) \\ x &= e^{-W(-\ln{2})} \\ x &= \frac{1}{e^{W(-\ln{2})}} \\ x &\approx -0.37927 - 0.72087 i\end{align}$$where W is the Lambert W function that I've written about in previous posts. See The Omega Constant and the Lambert W Function (June 24th 2020) and More on the Lambert W Function (February 16th 2021). To evaluate \(x\) using Wolfram Alpha the command ProductLog[-ln(2)] needs to be used.

Revisiting Odds And Evens

It's been a while since I posted about so-called odds and evens by which I mean the process of generating a new number from an existing one by adding its sum of odd digits and subtracting it's sum of even digits. An an example, let's consider 1234:$$1234 \rightarrow 1234+1+3-2-4=1232$$As we continue this process interesting things happen and that's what I wrote about in a paper that I published to Academia (link). Here are links to posts I've made about the topic:

• Odds and Evens on June 17th 2021
• Attractors, Vortices and Captives on June 18th 2021
• Binary Odds and Evens on June 19th 2021
• Odds and Evens: Statistics on June 21st 2021

Of course, if the sums of the odd and even digits are the same, the number remains unchanged. These sorts of numbers are what I termed "attractors" because other numbers, wherein there is an imbalance of odds and evens, have either these as their termini or they enter "vortices" or loops. My diurnal age today, 27326, is one such attractor which is why I was reminded of them. $$27326 \rightarrow 27326 + 7 + 3-2-2-6=27326$$Until now, I wasn't aware that they had their own OEIS sequence but they do and it is OEIS A036301 (permalink):

A036301

Numbers whose sum of even digits and sum of odd digits are equal.

The initial members of the sequence are:

0, 112, 121, 134, 143, 156, 165, 178, 187, 211, 314, 336, 341, 358, 363, 385, 413, 431, 516, 538, 561, 583, 615, 633, 651, 718, 781, 817, 835, 853, 871, 1012, 1021, 1034, 1043, 1056, 1065, 1078, 1087, 1102, 1120, 1201, 1210, 1223, 1232, 1245, 1254, 1267, 1276, 1289, 1298

I also discovered a sequence connected to what I termed "vorticals". These are the numbers that comprise a vortex into which non-balanced numbers are sucked if they do not end up in an attractor. The first instance of a vortex is that involving the numbers 11 and 13:$$ \begin{align} 11 \rightarrow 11 + 1 + 1 &= 13\\13 \rightarrow 13 + 1 + 3 &=17\\ 17 \rightarrow 17+1+7 &=25\\25 \rightarrow 25+5-2 &=28\\28 \rightarrow 28-2-8 &=18\\18 \rightarrow 18+1 -8 &=11 \end{align}$$Thus 11, 13, 17, 18, 25 and 28 are all numbers that eventually return to themselves after repeated mappings involving the odds and evens recursion. In the case of these numbers, six repetitions are needed: $$11 \rightarrow 13 \rightarrow 17 \rightarrow 25 \rightarrow 28 \rightarrow 18 \rightarrow 11$$Numbers like these are included in OEIS A124176:

A124176

Consider the map \(f\) that sends \(m\) to \(m\) + (sum of odd digits of \(m\)) - (sum of even digits of \(m\)). Sequence gives numbers \(m\) such that \(f^k(m) \) = \(m\) for some \(k\).

This will include all the attractors, the numbers that aren't changed by the mapping and where \(k\)=1. The initial members are (permalink):

0, 11, 13, 17, 18, 25, 28, 54, 55, 64, 65, 112, 121, 134, 137, 143, 148, 155, 156, 165, 166, 173, 178, 184, 187, 198, 200, 209, 211, 216, 231, 233, 234, 237, 244, 245, 270, 275, 280, 285, 314, 336, 341, 358, 363, 385, 396, 402, 407, 410, 413, 429, 431, 432

All other numbers are what I termed "captives" because they are "captured" by either an attractor or a "vortex" that is comprised of "vorticals". The attractors and vorticals are all listed in OEIS A124176 and OEIS A036301 is a subset of this sequence because it contains only the attractors that are invariant under the mapping.

There are 2974 such numbers in the range up to 40,000. Here are the numbers from 27326 up to 40000 that belong in the sequence (for future reference):

27326, 27344, 27362, 27380, 27412, 27421, 27434, 27437, 27443, 27448, 27455, 27456, 27465, 27466, 27473, 27478, 27484, 27487, 27528, 27546, 27564, 27582, 27601, 27610, 27617, 27623, 27624, 27632, 27635, 27642, 27645, 27653, 27654, 27660, 27667, 27676, 27689, 27698, 27748, 27766, 27784, 27803, 27825, 27830, 27847, 27852, 27869, 27874, 27896, 27968, 27986, 27990, 28004, 28007, 28013, 28019, 28037, 28055, 28073, 28091, 28093, 28095, 28097, 28099, 28101, 28103, 28105, 28107, 28109, 28190, 28239, 28257, 28275, 28293, 28295, 28297, 28299, 28301, 28303, 28305, 28307, 28329, 28370, 28392, 28459, 28477, 28495, 28497, 28499, 28501, 28503, 28505, 28527, 28549, 28550, 28572, 28594, 28679, 28697, 28699, 28701, 28703, 28725, 28730, 28747, 28752, 28769, 28774, 28796, 28899, 28901, 28910, 28917, 28923, 28924, 28932, 28935, 28942, 28945, 28953, 28954, 28960, 28967, 28976, 28989, 28998, 29018, 29081, 29108, 29126, 29144, 29162, 29180, 29216, 29238, 29261, 29283, 29328, 29346, 29364, 29382, 29414, 29436, 29441, 29458, 29463, 29485, 29548, 29566, 29584, 29612, 29621, 29634, 29637, 29643, 29648, 29655, 29656, 29665, 29666, 29673, 29678, 29684, 29687, 29768, 29786, 29801, 29810, 29817, 29823, 29824, 29832, 29835, 29842, 29845, 29853, 29854, 29860, 29867, 29876, 29889, 29898, 29988, 30014, 30036, 30041, 30058, 30063, 30085, 30104, 30122, 30140, 30178, 30181, 30212, 30221, 30234, 30237, 30243, 30248, 30255, 30256, 30265, 30266, 30273, 30278, 30284, 30287, 30306, 30324, 30342, 30360, 30378, 30383, 30384, 30401, 30410, 30417, 30423, 30424, 30432, 30435, 30442, 30445, 30453, 30454, 30460, 30467, 30476, 30489, 30498, 30508, 30526, 30544, 30562, 30580, 30597, 30603, 30608, 30615, 30617, 30618, 30621, 30622, 30625, 30630, 30647, 30652, 30669, 30674, 30696, 30728, 30746, 30764, 30782, 30795, 30805, 30806, 30812, 30813, 30819, 30824, 30827, 30849, 30850, 30872, 30894, 30948, 30966, 30984, 31004, 31022, 31040, 31078, 31081, 31116, 31138, 31161, 31183, 31202, 31220, 31235, 31238, 31245, 31248, 31318, 31381, 31400, 31411, 31413, 31417, 31418, 31425, 31428, 31454, 31455, 31464, 31465, 31598, 31608, 31609, 31611, 31616, 31631, 31633, 31634, 31637, 31644, 31645, 31670, 31675, 31680, 31685, 31807, 31810, 31813, 31829, 31831, 31832, 31850, 31851, 31853, 31857, 31860, 31865, 32012, 32021, 32034, 32037, 32043, 32048, 32055, 32056, 32065, 32066, 32073, 32078, 32084, 32087, 32102, 32120, 32135, 32138, 32145, 32148, 32201, 32210, 32217, 32223, 32224, 32232, 32235, 32242, 32245, 32253, 32254, 32260, 32267, 32276, 32289, 32298, 32304, 32322, 32340, 32378, 32381, 32397, 32403, 32408, 32415, 32417, 32418, 32422, 32425, 32430, 32447, 32452, 32469, 32474, 32496, 32506, 32524, 32542, 32560, 32578, 32583, 32584, 32595, 32605, 32606, 32615, 32616, 32627, 32649, 32650, 32672, 32694, 32708, 32726, 32744, 32762, 32780, 32793, 32804, 32807, 32810, 32813, 32829, 32870, 32892, 32928, 32946, 32964, 32982, 33006, 33024, 33042, 33060, 33078, 33083, 33084, 33118, 33181, 33204, 33222, 33240, 33278, 33281, 33402, 33420, 33435, 33438, 33445, 33448, 33600, 33611, 33613, 33617, 33618, 33625, 33628, 33654, 33655, 33664, 33665, 33809, 33811, 33816, 33831, 33833, 33834, 33837, 33844, 33845, 33870, 33875, 33880, 33885, 34001, 34010, 34017, 34023, 34024, 34032, 34035, 34042, 34045, 34053, 34054, 34060, 34067, 34076, 34089, 34098, 34100, 34111, 34113, 34117, 34118, 34125, 34128, 34154, 34155, 34164, 34165, 34197, 34203, 34208, 34213, 34214, 34225, 34230, 34247, 34252, 34269, 34274, 34296, 34302, 34320, 34335, 34338, 34345, 34348, 34395, 34405, 34408, 34411, 34427, 34449, 34450, 34472, 34494, 34504, 34522, 34540, 34578, 34581, 34593, 34606, 34607, 34609, 34611, 34629, 34670, 34692, 34706, 34724, 34742, 34760, 34778, 34783, 34784, 34791, 34793, 34801, 34804, 34805, 34807, 34809, 34811, 34890, 34908, 34926, 34944, 34962, 34980, 35008, 35026, 35044, 35062, 35080, 35206, 35224, 35242, 35260, 35278, 35283, 35284, 35404, 35422, 35440, 35478, 35481, 35602, 35620, 35635, 35638, 35645, 35648, 35800, 35811, 35813, 35817, 35818, 35825, 35828, 35854, 35855, 35864, 35865, 36003, 36025, 36030, 36047, 36052, 36069, 36074, 36096, 36109, 36111, 36116, 36131, 36133, 36134, 36137, 36144, 36145, 36170, 36175, 36180, 36185, 36195, 36205, 36206, 36207, 36209, 36212, 36213, 36227, 36249, 36250, 36272, 36294, 36300, 36311, 36313, 36317, 36318, 36325, 36328, 36354, 36355, 36364, 36365, 36393, 36399, 36403, 36404, 36405, 36407, 36417, 36418, 36429, 36470, 36492, 36502, 36520, 36535, 36538, 36545, 36548, 36591, 36593, 36597, 36601, 36602, 36603, 36605, 36607, 36609, 36612, 36615, 36690, 36704, 36722, 36740, 36778, 36781, 36789, 36794, 36795, 36803, 36806, 36813, 36906, 36924, 36942, 36960, 36978, 36983, 36984, 37028, 37046, 37064, 37082, 37208, 37226, 37244, 37262, 37280, 37406, 37424, 37442, 37460, 37478, 37483, 37484, 37604, 37622, 37640, 37678, 37681, 37802, 37820, 37835, 37838, 37845, 37848, 37995, 38005, 38006, 38012, 38013, 38027, 38028, 38049, 38050, 38072, 38094, 38107, 38110, 38113, 38129, 38131, 38132, 38150, 38151, 38153, 38157, 38160, 38165, 38193, 38195, 38199, 38201, 38203, 38204, 38205, 38207, 38210, 38213, 38229, 38270, 38292, 38309, 38311, 38316, 38331, 38333, 38334, 38337, 38344, 38345, 38370, 38375, 38380, 38385, 38391, 38399, 38402, 38409, 38412, 38415, 38490, 38500, 38511, 38513, 38517, 38518, 38525, 38528, 38554, 38555, 38564, 38565, 38589, 38590, 38591, 38595, 38599, 38600, 38601, 38603, 38607, 38609, 38614, 38617, 38702, 38720, 38735, 38738, 38745, 38748, 38774, 38779, 38787, 38788, 38789, 38792, 38797, 38801, 38808, 38815, 38904, 38922, 38940, 38978, 38981, 39048, 39066, 39084, 39228, 39246, 39264, 39282, 39408, 39426, 39444, 39462, 39480, 39606, 39624, 39642, 39660, 39678, 39683, 39684, 39804, 39822, 39840, 39878, 39881

Of course, the choice of adding the odd digits and subtracting the even digits is quite arbitrary and it's perfectly acceptable to reverse this and in doing so generate a different sequence. Doing this we generate OEIS A124177:

A124177

Consider the map \(f\) that sends \(m\) to \(m\) + (sum of even digits of \(m\)) - (sum of odd digits of \(m\) ). Sequence gives numbers \(m\) such that \(f^k(m)\) = \(m\) for some \(k\).

The initial members of this sequence are:

0, 22, 26, 27, 34, 35, 44, 49, 52, 63, 66, 78, 79, 81, 88, 99, 104, 107, 108, 112, 115, 121, 126, 133, 134, 143, 144, 151, 156, 165, 178, 187, 211, 224, 229, 232, 233, 283, 290, 314, 336, 341, 358, 363, 385, 413, 431, 467, 470, 489, 492, 516, 538, 561, 583, 615

Let's look at the first member of the sequence, 22:$$ \begin{align} 22 \rightarrow 22 + 2 + 2 &=26\\26 \rightarrow 26+2+6 &=34\\34 \rightarrow 34 + 4 - 3 &= 35\\35 \rightarrow 35-3-5 &=27\\27 \rightarrow 27+2-7 &=22 \end{align}$$Thus after five steps we arrive back at 22:$$22 \rightarrow 26 \rightarrow 34 \rightarrow 35 \rightarrow 27 \rightarrow 22$$It can be noted that "attractors", those numbers invariant under the mapping, remain the same but the "vorticals", those numbers that form "vortices" or loops, are different. There are 2966 members of this sequence in the range up to 40,000. For future reference, here are the members from 27326 to 40000:

27326, 27344, 27362, 27380, 27408, 27412, 27415, 27421, 27426, 27433, 27434, 27443, 27444, 27451, 27456, 27465, 27478, 27487, 27528, 27546, 27564, 27582, 27601, 27610, 27623, 27628, 27632, 27639, 27645, 27646, 27654, 27657, 27664, 27667, 27675, 27676, 27682, 27688, 27689, 27693, 27698, 27705, 27748, 27766, 27784, 27803, 27825, 27830, 27847, 27852, 27869, 27874, 27884, 27886, 27896, 27899, 27903, 27968, 27986, 28019, 28037, 28055, 28073, 28091, 28109, 28190, 28239, 28257, 28275, 28293, 28307, 28329, 28370, 28392, 28459, 28477, 28495, 28505, 28527, 28549, 28550, 28572, 28594, 28679, 28697, 28703, 28725, 28730, 28747, 28752, 28769, 28774, 28784, 28796, 28799, 28899, 28901, 28910, 28923, 28928, 28932, 28939, 28945, 28946, 28954, 28957, 28964, 28967, 28975, 28976, 28982, 28989, 28993, 28998, 29018, 29081, 29108, 29126, 29144, 29162, 29180, 29216, 29238, 29261, 29283, 29328, 29346, 29364, 29382, 29414, 29436, 29441, 29458, 29463, 29485, 29488, 29501, 29548, 29566, 29584, 29608, 29612, 29615, 29621, 29626, 29633, 29634, 29643, 29644, 29651, 29656, 29665, 29678, 29687, 29768, 29786, 29801, 29810, 29823, 29828, 29832, 29839, 29845, 29846, 29854, 29857, 29864, 29867, 29875, 29876, 29882, 29889, 29893, 29898, 29988, 30014, 30036, 30041, 30058, 30063, 30085, 30104, 30118, 30121, 30122, 30140, 30162, 30166, 30167, 30174, 30181, 30184, 30192, 30208, 30212, 30215, 30221, 30226, 30233, 30234, 30243, 30244, 30251, 30256, 30265, 30278, 30287, 30288, 30294, 30303, 30306, 30324, 30342, 30360, 30382, 30383, 30386, 30394, 30401, 30410, 30423, 30428, 30432, 30439, 30445, 30446, 30454, 30457, 30464, 30467, 30475, 30476, 30482, 30486, 30489, 30492, 30493, 30498, 30501, 30508, 30526, 30544, 30562, 30580, 30603, 30625, 30630, 30647, 30652, 30669, 30674, 30684, 30696, 30699, 30728, 30746, 30764, 30782, 30805, 30827, 30849, 30850, 30872, 30888, 30894, 30909, 30948, 30966, 30984, 31004, 31018, 31021, 31022, 31040, 31062, 31066, 31067, 31074, 31081, 31084, 31092, 31116, 31138, 31161, 31183, 31202, 31220, 31242, 31243, 31246, 31251, 31254, 31264, 31265, 31272, 31318, 31381, 31400, 31422, 31426, 31427, 31434, 31435, 31444, 31449, 31452, 31463, 31466, 31478, 31479, 31481, 31488, 31499, 31504, 31611, 31624, 31629, 31632, 31633, 31683, 31690, 31813, 31831, 31867, 31870, 31889, 31892, 32008, 32012, 32015, 32021, 32026, 32033, 32034, 32043, 32044, 32051, 32056, 32065, 32078, 32087, 32102, 32120, 32142, 32143, 32146, 32151, 32154, 32164, 32165, 32172, 32201, 32210, 32223, 32228, 32232, 32239, 32245, 32246, 32254, 32257, 32264, 32267, 32275, 32276, 32282, 32289, 32293, 32298, 32304, 32318, 32321, 32322, 32340, 32362, 32366, 32367, 32374, 32381, 32384, 32392, 32403, 32425, 32430, 32447, 32452, 32469, 32474, 32484, 32488, 32492, 32494, 32496, 32499, 32506, 32507, 32524, 32542, 32560, 32582, 32583, 32586, 32594, 32605, 32627, 32649, 32650, 32672, 32686, 32690, 32692, 32694, 32696, 32698, 32702, 32705, 32708, 32726, 32744, 32762, 32780, 32807, 32829, 32870, 32888, 32892, 32898, 32899, 32904, 32911, 32928, 32946, 32964, 32982, 33006, 33024, 33042, 33060, 33082, 33083, 33086, 33094, 33118, 33181, 33204, 33218, 33221, 33222, 33240, 33262, 33266, 33267, 33274, 33281, 33284, 33292, 33402, 33420, 33442, 33443, 33446, 33451, 33454, 33464, 33465, 33472, 33600, 33622, 33626, 33627, 33634, 33635, 33644, 33649, 33652, 33663, 33666, 33678, 33679, 33681, 33688, 33695, 33704, 33811, 33824, 33829, 33832, 33833, 33883, 33890, 34001, 34010, 34023, 34028, 34032, 34039, 34045, 34046, 34054, 34057, 34064, 34067, 34075, 34076, 34082, 34089, 34093, 34098, 34100, 34122, 34126, 34127, 34134, 34135, 34144, 34149, 34152, 34163, 34166, 34178, 34179, 34203, 34225, 34230, 34247, 34252, 34269, 34274, 34284, 34296, 34298, 34299, 34300, 34302, 34320, 34342, 34343, 34346, 34351, 34354, 34364, 34365, 34372, 34405, 34427, 34449, 34450, 34472, 34494, 34496, 34498, 34500, 34502, 34504, 34518, 34521, 34522, 34540, 34562, 34566, 34567, 34574, 34581, 34584, 34592, 34607, 34629, 34670, 34692, 34694, 34696, 34698, 34700, 34702, 34704, 34706, 34724, 34742, 34760, 34782, 34783, 34786, 34794, 34809, 34890, 34892, 34894, 34896, 34898, 34900, 34902, 34904, 34906, 34908, 34926, 34944, 34962, 34980, 35008, 35026, 35044, 35062, 35080, 35206, 35224, 35242, 35260, 35282, 35283, 35286, 35294, 35404, 35418, 35421, 35422, 35440, 35462, 35466, 35467, 35474, 35481, 35484, 35492, 35602, 35620, 35642, 35643, 35646, 35651, 35654, 35664, 35665, 35672, 35800, 35822, 35826, 35827, 35834, 35835, 35844, 35849, 35852, 35863, 35866, 35878, 35879, 35881, 35888, 35891, 35904, 36003, 36025, 36030, 36047, 36052, 36069, 36074, 36084, 36096, 36099, 36111, 36124, 36129, 36132, 36133, 36183, 36190, 36205, 36227, 36249, 36250, 36272, 36294, 36300, 36322, 36326, 36327, 36334, 36335, 36344, 36349, 36352, 36363, 36366, 36378, 36379, 36407, 36429, 36470, 36492, 36498, 36502, 36504, 36506, 36507, 36510, 36520, 36542, 36543, 36546, 36551, 36554, 36564, 36565, 36572, 36609, 36690, 36696, 36700, 36702, 36704, 36718, 36721, 36722, 36740, 36762, 36766, 36767, 36774, 36781, 36784, 36792, 36894, 36900, 36906, 36924, 36942, 36960, 36982, 36983, 36986, 36994, 37028, 37046, 37064, 37082, 37208, 37226, 37244, 37262, 37280, 37406, 37424, 37442, 37460, 37482, 37483, 37486, 37494, 37604, 37618, 37621, 37622, 37640, 37662, 37666, 37667, 37674, 37681, 37684, 37692, 37802, 37820, 37842, 37843, 37846, 37851, 37854, 37864, 37865, 37872, 38005, 38027, 38049, 38050, 38072, 38094, 38113, 38131, 38167, 38170, 38189, 38192, 38207, 38229, 38270, 38292, 38311, 38324, 38329, 38332, 38333, 38383, 38390, 38409, 38490, 38500, 38522, 38526, 38527, 38534, 38535, 38544, 38549, 38552, 38563, 38566, 38578, 38579, 38698, 38702, 38707, 38708, 38714, 38715, 38720, 38742, 38743, 38746, 38751, 38754, 38764, 38765, 38772, 38896, 38904, 38906, 38908, 38909, 38912, 38918, 38921, 38922, 38940, 38962, 38966, 38967, 38974, 38981, 38984, 38988, 38992, 39000, 39048, 39066, 39084, 39228, 39246, 39264, 39282, 39408, 39426, 39444, 39462, 39480, 39606, 39624, 39642, 39660, 39682, 39683, 39686, 39694, 39804, 39818, 39821, 39822, 39840, 39862, 39866, 39867, 39874, 39881, 39884, 39892

THE ESUCARYS MAPPING

 I've written about the Collatz or \(3x+1\) mapping and similar mappings in earlier posts:

• The \(3x+1\) Problem on November 1st 2016
• The Collatz Conjecture Revisited on March 15th 2018
• The \(17x+1\) Map on March 18th 2018
• The PrimeLatz Conjecture on August 25th 2019
• Palindromic Cyclops Numbers on August 10th 2020

I wasn't aware that the Collatz conjecture is also referred to an the Syracuse problem.

The Syracuse problem, also known as the Collatz conjecture or the \(3n+1\) conjecture or Ulam conjecture, is a very simple problem of arithmetics that is still unsolved today. It can be stated as follows:

Syracuse problem: \(n \geq 1\) being an integer, repeat the following operations

• If the number is even then divide it by two
• If the number is odd then multiply it by 3 and add 1

Conjecture: This process always reaches the number 1

Today, in investigating the number properties of my diurnal age (26251), I discovered that it is a member of OEIS A129133:

A129133

Numbers whose trajectory under the Esucarys map ends at the fixed point 247.

The Esucarys sequence derives its name from a reversal of "Syracuse", with the generating rule being that for the Syracuse (\(3x+1\) or Collatz) sequence followed by a reversal. 247 is the only known fixed point of the Esucarys sequence. Very few numbers map to 247. The members of this sequence, up to 26251, are:

247, 1247, 1484, 2473, 4859, 5087, 5738, 7318, 7484, 9563, 9682, 9694, 9938, 11247, 12189, 12473, 14840, 14842, 15209, 15610, 16274, 16563, 16750, 16798, 17609, 19168, 20019, 21885, 24733, 26251

Here is a permalink to the SageMath algorithm to generate this sequence. As an example, in reaching 247, 26251 follows this trajectory:

26251, 45787, 263731, 491197, 2953741, 4221688, 4480112, 6500422, 1120523, 751633, 94522, 16274, 7318, 9563, 9682, 1484, 247

Here is a permalink to the SageMath algorithm that will generate this trajectory. Figure 1 shows a graph of this trajectory.

Figure 1