First Fours, Threes and Twos of a Kind

The first thing we notice about \( \textbf{27807}\) is its factorisation:$$27807 = 3 \times 13 \times 23 \times 31$$This number has four distinct prime factors and all of them contain the digit 3. The number indicates my diurnal age today. How often does this occur (that a number has four distinct prime factors and all of them contain the digit 3). Well, 27807 is the first such number. Here is the list of numbers with this property up to 100,000: 27807, 33189, 38571, 44733, 47541, 51987, 62049, 64077, 65481, 74451, 76479, 79143, 88257, 88881, 91977, 92391. Table 1 shows these numbers together with their factorisations.

Table 1

A natural question to ask is what about other digits? In the range up to 100,000, there are only four numbers with four distinct prime factors all of which contain the digit 1. These are 46189, 75361, 84227 and 99671. Table 2 shows these numbers together with their factorisations.

Table 2

Apart from the digits 1 and 3, there are no other numbers in the range up to 100,000 with four distinct prime factors each of which contain the same digit. Such numbers exist of course but they are larger than 100,000. Table 3 shows the results for all the digits from 0 to 9.

Table 3: permalink

Thus we see that 27807 is unique in that it is the smallest number with four distinct prime factors such that each factor contains the same digit at least once. I'm pleased that I spotted this as it is easy to miss. We can construct a similar table for sphenic numbers as can be seen in Table 4 where the fourth factor appearing in Table 3 is omitted.

Table 4: permalink

So we see that 897 is the smallest sphenic number whose three distinct factors contain the same digit at least once. While we're here we may as well show the results for semiprimes with two distinct prime factors as well. See Table 5.

Table 5: permalink

Thus 39 is the smallest semiprime with two distinct prime factors such that each factor contains the same digit at least once. If we didn't specify distinct prime factors then 4 = 2 x 2 would win out.

Revisiting Reverse and Add

Numbers \(n\) belonging to OEIS A063048 have the property that the Reverse and Add! trajectory of \(n\) (presumably) does not reach a palindrome and does not join the trajectory of any term \(m < n\). Up to 40,000 these numbers are:

196, 879, 1997, 7059, 10553, 10563, 10577, 10583, 10585, 10638, 10663, 10668, 10697, 10715, 10728, 10735, 10746, 10748, 10783, 10785, 10787, 10788, 10877, 10883, 10963, 10965, 10969, 10977, 10983, 10985, 12797, 12898, 13097, 13197, 13694, 14096, 14698, 15297, 15597, 18598, 18798, 19098, 20459, 30389, 30399, 30929, 30959, 30979

Now there are many more numbers that presumably do not reach a palindrome but they join the trajectories of the above numbers at some point and thus do not fulfil the \(m<n\) condition. These numbers belong to OEIS A023108: positive integers which apparently never result in a palindrome under repeated applications of the function A056964(\(x\)) = \(x\) + (\(x\) with digits reversed). 

\( \textbf{27806} \), my diurnal age today, is one such number. I tested it for 30,000 iterations and still no palindrome was found. Now 39996 is the 1750th member of the sequence and so these numbers represent 4.375% of the total numbers in the range. Some numbers coming up soon are 27812,  27837,  27847,  27866,  27896,  27906,  27912,  27914,  27956,  27964 and 27988. Between 27806 and 40,000 these numbers (which include all those in OEIS A063048) are:

27806,  27812,  27837,  27847,  27866,  27896,  27906,  27912,  27914,  27956,  27964,  27988,  28036,  28046,  28055,  28056,  28095,  28096,  28146,  28156,  28193,  28236,  28256,  28266,  28281,  28286,  28332,  28341,  28342,  28346,  28356,  28362,  28369,  28469,  28487,  28496,  28502,  28504,  28545,  28546,  28566,  28576,  28586,  28595,  28596,  28597,  28616,  28643,  28657,  28686,  28696,  28702,  28704,  28706,  28707,  28732,  28736,  28776,  28796,  28797,  28802,  28827,  28837,  28856,  28886,  28896,  28902,  28904,  28946,  28954,  28978,  29026,  29036,  29045,  29046,  29085,  29086,  29097,  29136,  29146,  29183,  29226,  29246,  29256,  29271,  29276,  29322,  29331,  29332,  29336,  29346,  29352,  29359,  29396,  29459,  29477,  29486,  29494,  29499,  29535,  29536,  29556,  29566,  29576,  29585,  29586,  29587,  29590,  29606,  29633,  29647,  29676,  29686,  29722,  29726,  29766,  29786,  29787,  29791,  29796,  29817,  29827,  29846,  29876,  29886,  29899,  29936,  29944,  29968,  29997,  30089,  30358,  30389,  30399,  30439,  30458,  30479,  30489,  30536,  30551,  30561,  30575,  30581,  30583,  30636,  30651,  30661,  30666,  30695,  30713,  30726,  30733,  30744,  30746,  30781,  30783,  30785,  30786,  30841,  30849,  30875,  30881,  30889,  30929,  30931,  30959,  30961,  30963,  30967,  30975,  30979,  30981,  30983,  31079,  31348,  31379,  31389,  31429,  31448,  31469,  31479,  31526,  31541,  31551,  31565,  31571,  31573,  31626,  31641,  31651,  31656,  31685,  31703,  31716,  31723,  31734,  31736,  31771,  31773,  31775,  31776,  31831,  31839,  31865,  31871,  31879,  31896,  31919,  31921,  31949,  31951,  31953,  31957,  31965,  31969,  31971,  31973,  32069,  32095,  32295,  32338,  32369,  32379,  32391,  32419,  32438,  32459,  32469,  32516,  32531,  32541,  32555,  32561,  32563,  32616,  32631,  32641,  32646,  32675,  32706,  32713,  32724,  32726,  32761,  32763,  32765,  32766,  32791,  32795,  32821,  32829,  32855,  32861,  32869,  32886,  32896,  32909,  32911,  32939,  32941,  32943,  32947,  32955,  32959,  32961,  32963,  32999,  33059,  33085,  33095,  33195,  33285,  33328,  33359,  33369,  33381,  33390,  33391,  33395,  33409,  33428,  33449,  33459,  33499,  33506,  33521,  33531,  33545,  33551,  33553,  33594,  33595,  33606,  33621,  33631,  33636,  33665,  33692,  33703,  33714,  33716,  33751,  33753,  33755,  33756,  33781,  33785,  33811,  33819,  33845,  33851,  33859,  33876,  33886,  33901,  33929,  33931,  33937,  33945,  33949,  33951,  33953,  33989,  33995,  34049,  34075,  34085,  34094,  34095,  34185,  34195,  34275,  34295,  34318,  34349,  34359,  34371,  34380,  34381,  34385,  34395,  34418,  34439,  34449,  34489,  34511,  34521,  34535,  34541,  34584,  34585,  34611,  34621,  34626,  34655,  34682,  34696,  34704,  34706,  34741,  34745,  34746,  34771,  34775,  34801,  34809,  34835,  34841,  34849,  34866,  34876,  34895,  34899,  34919,  34921,  34923,  34927,  34935,  34939,  34941,  34979,  34985,  34993,  35039,  35065,  35075,  35084,  35085,  35175,  35185,  35265,  35285,  35295,  35308,  35339,  35349,  35361,  35370,  35371,  35375,  35385,  35391,  35398,  35408,  35429,  35439,  35479,  35498,  35501,  35511,  35525,  35531,  35533,  35574,  35575,  35595,  35601,  35611,  35616,  35645,  35672,  35686,  35731,  35733,  35735,  35736,  35761,  35765,  35825,  35831,  35839,  35856,  35866,  35885,  35889,  35909,  35911,  35913,  35917,  35925,  35929,  35931,  35933,  35969,  35975,  35983,  35999,  36029,  36055,  36065,  36074,  36075,  36165,  36175,  36255,  36275,  36285,  36329,  36339,  36351,  36360,  36361,  36365,  36375,  36381,  36388,  36419,  36429,  36469,  36488,  36501,  36515,  36521,  36523,  36564,  36565,  36585,  36595,  36601,  36606,  36635,  36662,  36676,  36721,  36723,  36725,  36726,  36751,  36755,  36795,  36815,  36821,  36829,  36846,  36856,  36875,  36879,  36901,  36903,  36907,  36915,  36919,  36921,  36923,  36959,  36965,  36973,  36989,  36997,  37019,  37045,  37055,  37064,  37065,  37155,  37165,  37245,  37265,  37275,  37290,  37295,  37319,  37329,  37341,  37350,  37351,  37355,  37365,  37371,  37378,  37409,  37419,  37459,  37478,  37496,  37499,  37505,  37511,  37513,  37554,  37555,  37575,  37585,  37595,  37625,  37652,  37666,  37695,  37711,  37713,  37715,  37716,  37741,  37745,  37785,  37805,  37811,  37819,  37836,  37846,  37865,  37869,  37895,  37905,  37909,  37911,  37913,  37949,  37955,  37963,  37979,  37987,  37999,  38009,  38035,  38045,  38054,  38055,  38094,  38095,  38099,  38145,  38155,  38192,  38235,  38255,  38265,  38280,  38285,  38309,  38319,  38331,  38340,  38341,  38345,  38355,  38361,  38368,  38399,  38409,  38449,  38468,  38486,  38489,  38495,  38499,  38501,  38503,  38544,  38545,  38565,  38575,  38585,  38594,  38595,  38596,  38615,  38642,  38656,  38685,  38695,  38701,  38703,  38705,  38706,  38731,  38735,  38775,  38795,  38796,  38801,  38809,  38826,  38836,  38855,  38859,  38885,  38895,  38899,  38901,  38903,  38939,  38945,  38953,  38969,  38977,  38989,  39025,  39035,  39044,  39045,  39084,  39085,  39089,  39096,  39135,  39145,  39182,  39225,  39245,  39255,  39270,  39275,  39309,  39321,  39330,  39331,  39335,  39345,  39351,  39358,  39389,  39395,  39439,  39458,  39476,  39479,  39485,  39489,  39498,  39534,  39535,  39555,  39565,  39575,  39584,  39585,  39586,  39605,  39632,  39646,  39675,  39685,  39721,  39725,  39765,  39785,  39786,  39790,  39791,  39795,  39816,  39826,  39845,  39849,  39875,  39885,  39889,  39891,  39898,  39929,  39935,  39943,  39959,  39967,  39979,  39996

The 17x + 1 Map Revisited

It's been a while, over seven years in fact since I last mentioned the 17\(x\) + 1 map in an eponymous post in March of 2018. As I explained back then:

Having recently written yet again about the Collatz trajectory, I was pleasantly surprised today to come upon a more generalised version of it. It goes by the name of the P\(x\) + 1 map of which the Collatz trajectory is a specific example in which P = 3. The P\(x\) + 1 trajectory or map is an algorithm that states: 

If \(x\) is divisible by any prime < P then divide out these primes one at a time starting with the smallest; otherwise multiply \(x\) by P and add 1.

My number for today is 25186 and it appears as an entry in OEIS A057534 that states:

• a(\(n\) +1) = a(\(n\))/2 if 2 | a(\(n\))
• a(\(n\)+1) = a(\(n\)) / 3 if 3 | a(\(n\))
• a(\(n\)+1) = a(\(n\)) / 5 if 5 | a(\(n\))
• a(\(n\)+1) = a(\(n\)) / 7 if 7 | a(\(n\))
• a(\(n\)+1) = a(\(n\)) / 11 if 11 | a(\(n\))
• a(\(n\)+1) = a(\(n\)) / 13 if 13 | a(\(n\))
• else a(\(n\)+1) = 17 x a(\(n\)) + 1

This is a particular example of the P\(x\) + 1 map in which P = 17 and this generates a sequence, part of which is shown below:

61, 1038, 519, 173, 2942, 1471, 25008, 12504, 6252, 3126, 1563, 521, 8858, 4429, 75294, 37647, 12549, 4183, 71112, 35556, 17778, 8889, 2963, 50372, 25186, 12593, 1799, 257, 4370, 2185, 437, 7430, 3715, 743, 12632, 6316, 3158, 1579, 26844, 13422, ...

Well, that was then, and so let's write out the above sequence in full because it is finite and loops. Here are the 84 terms (or 83 steps) with 61 added at the end to show the return to source:

61, 1038, 519, 173, 2942, 1471, 25008, 12504, 6252, 3126, 1563, 521, 8858, 4429, 75294, 37647, 12549, 4183, 71112, 35556, 17778, 8889, 2963, 50372, 25186, 12593, 1799, 257, 4370, 2185, 437, 7430, 3715, 743, 12632, 6316, 3158, 1579, 26844, 13422, 6711, 2237, 38030, 19015, 3803, 64652, 32326, 16163, 2309, 39254, 19627, 333660, 166830, 83415, 27805, 5561, 94538, 47269, 803574, 401787, 133929, 44643, 14881, 252978, 126489, 42163, 3833, 65162, 32581, 553878, 276939, 92313, 30771, 10257, 3419, 263, 4472, 2236, 1118, 559, 43, 732, 366, 183, 61

Why am I discussing this sequence again today? Well, the number associated with my diurnal age today (\( \textbf{27805} \)) is a member of this sequence. We can see this more clearly if we arrange the terms in ascending order.

43, 61, 173, 183, 257, 263, 366, 437, 519, 521, 559, 732, 743, 1038, 1118, 1471, 1563, 1579, 1799, 2185, 2236, 2237, 2309, 2942, 2963, 3126, 3158, 3419, 3715, 3803, 3833, 4183, 4370, 4429, 4472, 5561, 6252, 6316, 6711, 7430, 8858, 8889, 10257, 12504, 12549, 12593, 12632, 13422, 14881, 16163, 17778, 19015, 19627, 25008, 25186, 26844, \( \textbf{27805} \), 30771, 32326, 32581, 35556, 37647, 38030, 39254, 42163, 44643, 47269, 50372, 64652, 65162, 71112, 75294, 83415, 92313, 94538, 126489, 133929, 166830, 252978, 276939, 333660, 401787, 553878, 803574

We can see that after 25186, the number that prompted my original post, there has only been one other member (26844) until today. If we start with 27805 then the sequence returns to this same number after 85 iterations (permalink):

27805, 5561, 94538, 47269, 803574, 401787, 133929, 44643, 14881, 252978, 126489, 42163, 3833, 65162, 32581, 553878, 276939, 92313, 30771, 10257, 3419, 263, 4472, 2236, 1118, 559, 43, 732, 366, 183, 61, 1038, 519, 173, 2942, 1471, 25008, 12504, 6252, 3126, 1563, 521, 8858, 4429, 75294, 37647, 12549, 4183, 71112, 35556, 17778, 8889, 2963, 50372, 25186, 12593, 1799, 257, 4370, 2185, 437, 7430, 3715, 743, 12632, 6316, 3158, 1579, 26844, 13422, 6711, 2237, 38030, 19015, 3803, 64652, 32326, 16163, 2309, 39254, 19627, 333660, 166830, 83415, 27805

Figure 1 shows a plot of these values using a logarithmic scale:

Figure 1: permalink

Now this sequence of length 84 or 83 steps is not the longest by far, In fact as the numbers get larger then the lengths of the record breaking sequence lengths also increases steadily. Table 1 shows these record step lengths for numbers up to 40000 as well as indicating whether the sequences end up looping or reaching 1.

\( \textbf{17x + 1} \)

Table 1: record step lengths of 17\(x\) + 1
permalink

It must be said the generation of this table timed out using SageMathCell and needed to be completed in my Jupyter notebook. While we are on the topic, we should look at the record step lengths for primes 3, 5, 7, 11 and 13 as well. Let's start with 13\(x\) + 1. The numbers with record breaking steps are as follows (with Table 2 showing more detail):

1, 2, 4, 8, 13, 26, 41, 61, 122, 197, 271, 529, 661, 1322, 1607, 3214, 4337, 4597, 4981, 7663, 15326, 27859

\( \textbf{13x + 1} \)

Table 2: record step lengths of 13\(x\) + 1
permalink

The numbers with record breaking steps for P11 + 1 are as follows (with Table 3 showing more detail):

1, 2, 4, 8, 11, 22, 23, 46, 92, 151, 247, 407, 653, 883, 977, 1313, 1703, 2477, 4954, 6847, 12449, 14471, 19013, 21527, 22627, 39281

\( \textbf{11x + 1} \)

Table 3: record steps lengths of 11\(x\) + 1
permalink

The numbers with record steps for P7 + 1 are (with Table 4 showing more details):

1, 3, 6, 7, 11, 19, 31, 49, 53, 106, 121, 163, 283, 343, 403, 806, 1471, 1681, 1919, 3133, 4243, 4849, 8659, 11683, 12373, 24746, 30301, 35803

\( \textbf{7x + 1} \)

Table 4: record step lengths of 7\(x\) + 1
permalink

The numbers with record steps for P5+1 are (with Table 5 showing more details):

1, 2, 4, 5, 10, 20, 23, 46, 47, 85, 95, 190, 380, 383, 766, 919, 1655, 2117, 3575, 6097, 6503, 10463, 12053, 24106, 28927, 39053

\( \textbf{5x + 1} \)

Table 5: record steps lengths of 5\(x\) + 1
permalink

The numbers with record lengths for P3 + 1 (with Table 6 showing more details) are:

1, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, 2223, 2463, 2919, 3711, 6171, 10971, 13255, 17647, 23529, 26623, 34239, 35655

\( \textbf{3x + 1} \)

Table 6: record lengths for 3\(x\) + 1
permalink

If we collect all the numbers from all the sequences above then we have a list of all the numbers that reach a record number of steps under the P\(x\) + 1 mappings where \(x\) = 3, 5, 7, 11, 13 and 17. Here is the list in the range from 1 to 40,000:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 18, 19, 20, 22, 23, 25, 26, 27, 31, 41, 43, 46, 47, 49, 53, 54, 61, 73, 85, 92, 95, 97, 106, 121, 122, 129, 151, 163, 171, 173, 183, 190, 197, 231, 247, 257, 263, 271, 283, 313, 327, 343, 366, 380, 383, 403, 407, 437, 519, 521, 529, 559, 649, 653, 661, 703, 732, 743, 766, 806, 871, 883, 919, 977, 1038, 1118, 1161, 1313, 1322, 1471, 1563, 1579, 1607, 1655, 1681, 1703, 1799, 1919, 2117, 2185, 2223, 2236, 2237, 2309, 2463, 2477, 2919, 2942, 2963, 3126, 3133, 3158, 3214, 3419, 3575, 3711, 3715, 3803, 3833, 4183, 4243, 4337, 4370, 4429, 4472, 4597, 4849, 4954, 4981, 5561, 6097, 6171, 6252, 6316, 6503, 6711, 6847, 7430, 7663, 8659, 8858, 8889, 10257, 10463, 10971, 11683, 12053, 12373, 12449, 12504, 12549, 12593, 12632, 13255, 13422, 14471, 14881, 15326, 16163, 17647, 17778, 19013, 19015, 19627, 21527, 22627, 23529, 24106, 24746, 25008, 25186, 26623, 26844, 27805, 27859, 28927, 30301, 30771, 32326, 32581, 34239, 35556, 35655, 35803, 37647, 38030, 39053, 39254, 39281

Revisiting Fibodiv and Repfigit Numbers

On the 21st January 2024, I made a post titled Fibodiv Numbers and today I'm revisiting this topic because \( \textbf{27801} \), the number associated with my diurnal age today, is one such number. There aren't many of them. In the range up to one million, there are only 233. Up to 40,000 these are (permalink):

14, 19, 28, 47, 61, 75, 122, 149, 183, 199, 244, 298, 305, 323, 366, 427, 488, 497, 549, 646, 795, 911, 969, 1292, 1301, 1499, 1822, 1999, 2087, 2602, 2733, 2998, 3089, 3248, 3379, 3644, 3903, 4555, 4997, 5204, 5466, 6178, 6377, 6496, 6505, 7288, 7806, 7995, 8199, 8845, 9107, 9161, 9267, 9744, 10408, 11709, 12356, 12992, 13010, 14311, 14999, 15445, 15612, 16913, 17690, 18214, 18322, 18534, 19515, 19999, 20816, 20987, 21623, 22117, 23418, 24712, 24719, 26020, 27321, 27483, 27801, 28622, 29107, 29923, 29998, 30890, 31224, 32498, 32525, 33826, 33979, 35127, 36428, 36644, 37729, 39030

These numbers form OEIS A130792: numbers \(k\) whose representation can be split in two parts which can be used as seeds for a Fibonacci-like sequence containing \(k\) itself. 

In that OEIS entry Mathematica code has been entered which is incomprehensible to me but fortunately Google Gemini (or similar) can be used to convert this code to Python that can then be run in SageMathCell. The Python code runs perfectly and quickly generates the 233 Fibodiv numbers. Gemini also provides an explanation of how the code works and additional help can be obtained if needed (link). This ease of converting from any programming language to a language of ones own choice is very useful.

So let's see how 27801 earns it right of inclusion into OEIS A130792 (permalink):$$27, 801, 828, 1629, 2457, 4086, 6543, 10629, 17172, 27801$$We see how the two parts of 27801, 27 and 801, serve as seeds for a Fibonacci-like sequence that eventually generates the number 27801.

Now Repfigit numbers are similar to Fibodiv numbers and in fact the two digit Repfigit numbers are also Fibodiv numbers. I discussed these in my post On Turning 75 on April 3rd 2024. They belong to OEIS A007629: Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers): numbers \(n\) with \(k\) digits such that a Fibonacci-like sequence can be defined using as seeds the digits of \(n\) and then at each step adding the last \(k\) terms. If \(n\) itself appears in the sequence, then it is a repfigit number. Up to 40000, the members of this sequence are (permalink):

14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909, 31331, 34285, 34348

An example is 197 with three digits such that:$$1, 9, 7, 17, 33, 57, 107, 197$$We can see that:$$ \begin{align} 1 + 9 + 7 &= 17\\ 9+7+17 &= 33 \\ 7 + 17 + 33 &=57 \\ 17 + 33 + 57 &=107 \\33+57+107 &= 197 \end{align} $$A full list of Keith numbers can be found at this site. 

Some Interesting Integer Ratios

The number \( \textbf{27800} \) (my diurnal age today) has the following divisors: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 139, 200, 278, 556, 695, 1112, 1390, 2780, 3475, 5560, 6950, 13900, 27800. If we concatenate these divisors in the order shown (from smallest to largest) we get the rather large integer shown below.

1245810202540501001392002785566951112139027803475556069501390027800

The sum of these divisors is 65100 and it so happens that 65100 divides this concatenated number without remainder to give:

19136869470668218147342592712241952567419781927427896612924578

Numbers with this property belong to OEIS A308486: numbers such that the sum of divisors divides the concatenation (in ascending order) of divisors. The initial members up to 40000 are (permalink):

1, 2, 6, 10, 40, 98, 112, 120, 1904, 2680, 4040, 4128, 5136, 9920, 12224, 17900, 20880, 27800

Looking at the number 98 in the list we see that its divisors are 1, 2, 7, 14, 49 and 98 that have a total sum of 171 and whose concatenated divisors form the number 127144998. Thus we have:$$ \begin{align} \frac{127144998}{171} &= \frac{2 \times 3^3 \times 19 \times 123923}{3^2 \times 19} \\ &= 2 \times 3 \times 123932 \\ &=743538 \end{align} $$What if we consider the concatenation of a number's factors (with repetition) and whether it can be divided by its sum of factors (again with repetition). We need to ignore the prime numbers or else they will all get included. There are 277 composite numbers satisfying the criterion in the range up to 40000 (permalink):

8, 14, 20, 24, 27, 62, 125, 150, 160, 180, 194, 218, 300, 343, 452, 510, 512, 548, 570, 605, 612, 627, 651, 662, 663, 720, 935, 1183, 1210, 1235, 1331, 1335, 1575, 1676, 1994, 2090, 2106, 2130, 2197, 2218, 2303, 2337, 2345, 2350, 2428, 2436, 2640, 2667, 2675, 2679, 2744, 3087, 3102, 3108, 3168, 3237, 3275, 3399, 3509, 3553, 3740, 3835, 4029, 4046, 4125, 4180, 4347, 4384, 4392, 4410, 4488, 4565, 4704, 4805, 4913, 5015, 5037, 5047, 5120, 5551, 5829, 5888, 5968, 6223, 6250, 6549, 6662, 6666, 6747, 6837, 6859, 6888, 6923, 7030, 7189, 7337, 7448, 7449, 7462, 7488, 8000, 8064, 8165, 8246, 8421, 8624, 8742, 8853, 8949, 9000, 9331, 9344, 9709, 9804, 9975, 9990, 10017, 10125, 10146, 10240, 10387, 10800, 10854, 10865, 10879, 10989, 11045, 11121, 11205, 11264, 11704, 11891, 12032, 12152, 12167, 12288, 12337, 13237, 13243, 13277, 13284, 13702, 13792, 13824, 13858, 14308, 14457, 14555, 14580, 15015, 15025, 15042, 15054, 15301, 15552, 15820, 16038, 16428, 16549, 16827, 16856, 17347, 17496, 17600, 17850, 17914, 17949, 18172, 18213, 18377, 18495, 18821, 18963, 19135, 19425, 19513, 19683, 19860, 19885, 19910, 20041, 20083, 20727, 20746, 20878, 20951, 21033, 21175, 21197, 21340, 21965, 21978, 22008, 22021, 22152, 22275, 22317, 23069, 23280, 23548, 23715, 23785, 23998, 24037, 24244, 24389, 24986, 25182, 25344, 25647, 26129, 26754, 27010, 27480, 27664, 27832, 27880, 28006, 28037, 28566, 28577, 28840, 28896, 29064, 29281, 29326, 29388, 29602, 29614, 29624, 29783, 29791, 30082, 30186, 30226, 30229, 30240, 30420, 30814, 30825, 31097, 31349, 31412, 31581, 31780, 32076, 32418, 32640, 32697, 33292, 33473, 33480, 34132, 34133, 34481, 34521, 34773, 35046, 35557, 35616, 36022, 36040, 36162, 36176, 36478, 36504, 37026, 37789, 38024, 38200, 38340, 38399, 38480, 38658, 39292, 39406, 39463

This sequence of numbers is NOT to be found in the OEIS. Let's look at one of the numbers in the above list, namely 27832.$$27832=2^3 \times 7^2 \times 71$$The concatenated factors form the number 2227771 and the sum of these divisors is 91. Thus we have:$$ \begin{align} \frac{2227771}{91} &= \frac{7 \times 13 \times 24481}{7 \times 13} \\ &=24481 \end{align} $$There are other variations on the two themes covered in this post. For example, we could consider only the proper divisors of a number and look for numbers whose proper divisors, when concatenated from smallest to largest, are divisible by the sum of the proper divisors. We need to exclude prime numbers because the proper divisor in every case is 1 and thus will divide any number. There are 38 composite numbers in the range up to 40000 (permalink):

4, 15, 18, 24, 69, 208, 247, 501, 559, 565, 692, 697, 1501, 2077, 2257, 2759, 3551, 3661, 4135, 4227, 5123, 5461, 5536, 6109, 8640, 10821, 12179, 12667, 13631, 16939, 19781, 23587, 24307, 26827, 27331, 30701, 33877, 38887

Let's consider 69 in the previous list. It has proper divisors of 1, 3 and 23 that form the concatenated number 1323 with a sum of 27. Thus we have:$$ \begin{align} \frac{1323}{27} &= \frac{3^3 \times 7^2}{3^3} \\ &=7^2 \\ &=49 \end{align}$$Again this sequence is NOT to be found in the OEIS. Interestingly, the concatenated proper divisors of 8640 (one of the sequence members) form the following enormous cancatenated number:

12345689101215161820242730323640454854606472809096108120135144160180192216240270288320360432480540576720864960108014401728216028804320

An Interesting Sequence of Primes

Consider the sequence of primes and the products of their digits (POD). As the primes increase, these products will reach record values as certain primes are encountered. Table 1 shows these primes above 20000 together with their record products (permalink):

Table 1

Here are the primes without their products (listed in OEIS A230041):

2, 3, 5, 7, 19, 29, 37, 47, 59, 79, 89, 199, 269, 359, 379, 389, 479, 499, 599, 797, 887, 997, 1889, 1999, 2689, 2699, 2789, 2999, 3889, 3989, 4789, 4799, 4889, 4999, 6899, 8999, 25999, 27799, 28789, 28979, 29989, 37799, 37889, 39799, 39989

\( \textbf{27799} \) is the number associated with my diurnal age today which is why I was alerted to this sequence. If we consider both prime and non-prime numbers then the results are shown in Table 2 for numbers greater than 20000.

Table 2

Here is a list of the numbers with record breaking products up to 40000 (see OEIS A095706):

2, 3, 4, 5, 6, 7, 8, 9, 25, 26, 27, 28, 29, 37, 38, 39, 47, 48, 49, 58, 59, 68, 69, 78, 79, 88, 89, 99, 259, 268, 269, 278, 279, 288, 289, 299, 378, 379, 388, 389, 399, 479, 488, 489, 499, 589, 599, 689, 699, 789, 799, 889, 899, 999, 2599, 2689, 2699, 2789, 2799, 2889, 2899, 2999, 3789, 3799, 3889, 3899, 3999, 4799, 4889, 4899, 4999, 5899, 5999, 6899, 6999, 7899, 7999, 8899, 8999, 9999, 25999, 26899, 26999, 27899, 27999, 28899, 28999, 29999, 37899, 37999, 38899, 38999, 39999

An Interesting Looping Digit Sequence

A paucity of information about a number always challenges me to find something interesting about the number. Such was the case with the number associated with my diurnal age today: \( \textbf{27796} \). I could see that the first two digits of the number, 2 and 7, added to give the fourth digit and the fourth digit plus the fifth digit gave the rightmost digit (6) of the resultant sum (16). This was almost a Fibonacci sequence mod 10 except that the repeated digit 7 got in the way. 

So I devised the following set of rules. 

1. Let \(a\) and \(b\) be the first two digits of the sequence \( (a,b)\)
2. If \(b \neq 7 \), \(c=a+b \pmod {10}\) gives the next digit
3. Let \(a,b=b,c\)
4. If \(b=7\) then \(c=7\) and \(d=a +b \pmod {10} \) give the next two digits
5. Let \(a,b=c,d\)

Applying these rules when \(a=2\) and \(b=7\) gives the following 68 member sequence (permalink):

[2, 7, 7, 9, 6, 5, 1, 6, 7, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 7, 2, 9, 1, 0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 7, 0, 7, 7, 7, 7, 4, 1, 5, 6, 1, 7, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 7, 5]

How might we define 27796 in terms of this sequence? Well, perhaps like this. 27796 is the concatenation of the first five digits of the previous sequence created according to the rules 1 to 5 defined above. The digits of the sequence repeat when the 69th term is reached. The final two digits, 7 and 5, mark the end of the sequence because:$$ \begin{align} 7 + 5 &= 2 \pmod {10} \\ 5 + 2 &= 7 \end{align} $$Thus we have the same two digits, 2 and 7, that we started with. Figure 1 shows the progression of digits.

Figure 1: permalink

It can be considered that we are building a 68 digit number from a starting two digit number of 27. We might represent the number thus (with the superscript 7 representing the digit that is repeated and the overline indicating the cycle of 68 digits):$$ ^7 \, \overline{27796 \dots 98775}$$The number builds as follows:$$27, 277, 2779, 27796, \dots$$If we start with \(a=0\) and \(b=1\), as in the classic Fibonacci sequence, we still end up with a cycle of 68 and 7 is still the digit that's repeated. Here is the sequence (permalink) with Figure 2 showing the progression in graphical format:

[0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 7, 0, 7, 7, 7, 7, 4, 1, 5, 6, 1, 7, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 7, 5, 2, 7, 7, 9, 6, 5, 1, 6, 7, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 7, 2, 9, 1, 0]

Figure 2: permalink

It's easy to modify the algorithm so that a digit other than 7 gets repeated. Let's repeat the digit 6. In this case we end up with following 64 member looping sequence (permalink) with Figure 3 showing the progression in graphical format:

[2, 7, 9, 6, 6, 5, 1, 6, 6, 7, 3, 0, 3, 3, 6, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1, 0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5]

Figure 3: permalink

This time we can consider that we are building a 64 digit number:$$ ^6 \, \overline{27966 \dots 99875}$$The number builds as follows:$$27, 279, 2796, 27966, \dots$$Thus 27966 could similarly be described as the concatenation of the first five digits of the previous sequence created according to the rules 1 to 5 defined above. The digits of the sequence repeat when the 65th term is reached. There are many other combinations of digits that could be explored but the point of this post is that interesting number patterns can always be found with just a little scrutiny and imagination.

I've made two previous posts about numbers whose digits display Fibonacci-like properties. These posts are Additive Fibonacci-like Numbers in August 2024 and Consolidating Fibonacci-like Numbers in November of 2024. In the former post I looked at numbers like 28191 where the arithmetic digital root was invoked to reveal a Fibonacci-like progression:$$ \begin{align} 2+8 &=10 \rightarrow 1 \\8+1 &= 9 \\1+9 &=10 \rightarrow 1  \end{align}$$Here the digital root of the two previous digits determines the next digit which is then concatenated with the previous digits. This process again produces a looping sequence of digits and a 24 digit number that repeats endlessly (permalink):$$ \begin {align} &2, 8, 1, 9, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6\\\ &\overline{281911235843718988764156} \end{align}$$Alternatively, the digital root of all the previous digits can be used to determine the next digit. For example, 26875 where:$$ \begin{align} 2 + 6 &= 8\\ 2+6 + 8 &=16 \rightarrow 7\\ 2 + 6 + 8 + 7 &=23 \rightarrow 5 \end{align}$$In Consolidating Fibonacci-like Numbers, I considered numbers in base 10 that display Fibonacci-like properties when converted to other bases. I did not use digital roots for these cases, only digit sums. For example, 27802 in base 14:$$ \begin{align} 27802 &=\text{a1bc}_{14}\\ \text{a+1}&=\text{b} \\ \text{1+b} &=\text{c} \end{align}$$Related: The Pisano Period.