Semiprime Chains

The number associated with my diurnal age today is 27578 and it is a squarefree semiprime with an interesting property. Let's consider its two factors and subtract the smaller from the larger factor and apply the same rule to the difference. Keep repeating this process until the difference is not a squarefree semiprime. The result is as follows:$$ \begin{align} 27578 &= 2 \times 13789\\ 13789-2 &= 13787 \\13787 &= 17 \times 811\\811-17 &=794 \\794 &= 2 \times 397\\397-2 &= 395 \\ 395 &= 5 \times 79 \\79 -5 &= 74 \\ 74 &= 2 \times 37\\37-2 &= 35\\35 &= 5 \times 7 \end{align}$$Once we reach 35, the chain of semiprimes terminates because the difference between 7 and 5 is 2 and 2 is not a squarefree semiprime. However, the process does generate a chain of semiprimes:$$27578 \rightarrow 13787 \rightarrow 794 \rightarrow 395 \rightarrow 74 \rightarrow 35$$Squarefree semiprimes like 27578 that produce another five squarefree semiprimes by subtraction of their prime factors belong to OEIS A296812:

A296812    Take a squarefree semiprime and take the difference of its prime factors. If it is a squarefree semiprime repeat the process. Sequence lists the squarefree semiprimes that generate other squarefree semiprimes only in the first \(k\) steps of this process. Case \(k \geq 5\).

The initial members of this sequence, up to 40000, are (permalink):

4786, 5991, 6218, 8351, 9995, 13391, 14367, 15434, 16658, 16706, 18663, 19466, 27578, 28738, 33551, 34082, 34187, 37727, 38823

The numbers marked in red correspond to the case where \(k \geq 6\), although these numbers are not listed in the OEIS. Take 33551 as an example:$$ \begin{align} 33551 &= 7 \times 4793\\4793 - 7 &= 4786\\4786 &= 2 \times 2393\\2393-2 &= 2391\\2391 &= 3 \times 797\\797-3 &= 794\\794 &= 2 \times 397\\397-2 &= 395\\ 395 &= 5 \times 79\\ 79 - 5 &=74\\74 &= 2 \times 37\\37-2 &=35\\35 &= 5 \times 7 \end{align}$$The number in blue in the list above (28738) corresponds to the case where \(k\)=7 and the prime factors of this number are 2 x 14369 which leads us to 14367 (one of the red numbers).

A similar process involving addition of the prime factors could be applied this would lead, in the case of \(k \geq 5\), to this sequence of numbers:

1774, 2566, 2913, 4497, 6382, 6769, 8902, 9286, 10334, 15177, 19357, 28177, 34669, 35913, 37857

Take 1774 as an example where we have:$$ \begin{align} 1774 &= 2 \times 887\\ 887+2 &= 889\\889 &= 7 \times 127\\127+7 &=134\\134 &= 2 \times 67\\67+2 &= 69\\ 69 &= 3 \times 23\\23+3 &= 26\\26 &= 2 \times 13\\13+2 &=15\\15 &= 3 \times 5 \end{align} $$These numbers are NOT listed in the OEIS. I'm sure some of these numbers could be taken further as with the differences but I'll leave it there for now.

Katadromes

Having dealt with metadromes in an earlier post, it's time to look at katadromes the opposite of these numbers. According to Numbers Aplenty (which site is now working again):

A number is a katadrome in a given base \(b\)  (often 10 or 16) if its digits are in strictly decreasing order in that base. For example, 43210, 76521 and 9630 are all katadromes in base 10. If we allow the digits of a katadrome to be non-strictly decreasing (i.e., nonincreasing, like in 43310 or 2222, we obtain nialpdromes. Similarly, the numbers whose digits are nondecreasing and strictly increasing are called plaindromes and metadromes, respectively. The total number katadromes in base \(b\)  is equal to \(2^b-1\), hence in base 10 there are \(2^{10}-1 = 1034 - 1 =1023\) katadromes, from 0 to 9876543210.

Here are the katadromes between 9000 and 90000 (link):

..., 9210, 9310, 9320, 9321, 9410, 9420, 9421, 9430, 9431, 9432, 9510, 9520, 9521, 9530, 9531, 9532, 9540, 9541, 9542, 9543, 9610, 9620, 9621, 9630, 9631, 9632, 9640, 9641, 9642, 9643, 9650, 9651, 9652, 9653, 9654, 9710, 9720, 9721, 9730, 9731, 9732, 9740, 9741, 9742, 9743, 9750, 9751, 9752, 9753, 9754, 9760, 9761, 9762, 9763, 9764, 9765, 9810, 9820, 9821, 9830, 9831, 9832, 9840, 9841, 9842, 9843, 9850, 9851, 9852, 9853, 9854, 9860, 9861, 9862, 9863, 9864, 9865, 9870, 9871, 9872, 9873, 9874, 9875, 9876, 43210, 53210, 54210, 54310, 54320, 54321, 63210, 64210, 64310, 64320, 64321, 65210, 65310, 65320, 65321, 65410, 65420, 65421, 65430, 65431, 65432, 73210, 74210, 74310, 74320, 74321, 75210, 75310, 75320, 75321, 75410, 75420, 75421, 75430, 75431, 75432, 76210, 76310, 76320, 76321, 76410, 76420, 76421, 76430, 76431, 76432, 76510, 76520, 76521, 76530, 76531, 76532, 76540, 76541, 76542, 76543, 83210, 84210, 84310, 84320, 84321, 85210, 85310, 85320, 85321, 85410, 85420, 85421, 85430, 85431, 85432, 86210, 86310, 86320, 86321, 86410, 86420, 86421, 86430, 86431, 86432, 86510, 86520, 86521, 86530, 86531, 86532, 86540, 86541, 86542, 86543, 87210, 87310, 87320, 87321, 87410, 87420, 87421, 87430, 87431, 87432, 87510, 87520, 87521, 87530, 87531, 87532, 87540, 87541, 87542, 87543, 87610, 87620, 87621, 87630, 87631, 87632, 87640, 87641, 87642, 87643, 87650, 87651, 87652, 87653, 87654, ...

In terms of my diurnal age it can be seen that 9876 was the last number associated with that and then there is the huge jump to 43210 that I'll never get to experience. An interesting fact is that :$$p_{8510}=87641$$is the largest katadromic prime whose index is a katadromic too.

However,  if we consider katadromes in base 16, then the decimal equivalents of these katadromes form OEIS A023797:

 A023797: Katadromes: digits in base 16 are in strict descending order.

Here are the decimal equivalents from 29000 to 40000 (permalink to Python code):

..., 29200, 29456, 29472, 29473, 29712, 29728, 29729, 29744, 29745, 29746, 29968, 29984, 29985, 30000, 30001, 30002, 30016, 30017, 30018, 30019, 30224, 30240, 30241, 30256, 30257, 30258, 30272, 30273, 30274, 30275, 30288, 30289, 30290, 30291, 30292, 33296, 33552, 33568, 33569, 33808, 33824, 33825, 33840, 33841, 33842, 34064, 34080, 34081, 34096, 34097, 34098, 34112, 34113, 34114, 34115, 34320, 34336, 34337, 34352, 34353, 34354, 34368, 34369, 34370, 34371, 34384, 34385, 34386, 34387, 34388, 34576, 34592, 34593, 34608, 34609, 34610, 34624, 34625, 34626, 34627, 34640, 34641, 34642, 34643, 34644, 34656, 34657, 34658, 34659, 34660, 34661, 37392, 37648, 37664, 37665, 37904, 37920, 37921, 37936, 37937, 37938, 38160, 38176, 38177, 38192, 38193, 38194, 38208, 38209, 38210, 38211, 38416, 38432, 38433, 38448, 38449, 38450, 38464, 38465, 38466, 38467, 38480, 38481, 38482, 38483, 38484, 38672, 38688, 38689, 38704, 38705, 38706, 38720, 38721, 38722, 38723, 38736, 38737, 38738, 38739, 38740, 38752, 38753, 38754, 38755, 38756, 38757, 38928, 38944, 38945, 38960, 38961, 38962, 38976, 38977, 38978, 38979, 38992, 38993, 38994, 38995, 38996, 39008, 39009, 39010, 39011, 39012, 39013, 39024, 39025, 39026, 39027, 39028, 39029, 39030, ...

As can be seen, given my current diurnal age of 27576 days, it will some time before I enjoy a katadromic day in base 16 (29200 - 27576 = 1624 days away corresponding to 14th March 2029). I have to confess to not understanding the Python code used to generate these numbers. Here are the hexadecimal katadromes from 29000 (decimal) to 40000 (decimal) - permalink:

7210, 7310, 7320, 7321, 7410, 7420, 7421, 7430, 7431, 7432, 7510, 7520, 7521, 7530, 7531, 7532, 7540, 7541, 7542, 7543, 7610, 7620, 7621, 7630, 7631, 7632, 7640, 7641, 7642, 7643, 7650, 7651, 7652, 7653, 7654, 8210, 8310, 8320, 8321, 8410, 8420, 8421, 8430, 8431, 8432, 8510, 8520, 8521, 8530, 8531, 8532, 8540, 8541, 8542, 8543, 8610, 8620, 8621, 8630, 8631, 8632, 8640, 8641, 8642, 8643, 8650, 8651, 8652, 8653, 8654, 8710, 8720, 8721, 8730, 8731, 8732, 8740, 8741, 8742, 8743, 8750, 8751, 8752, 8753, 8754, 8760, 8761, 8762, 8763, 8764, 8765, 9210, 9310, 9320, 9321, 9410, 9420, 9421, 9430, 9431, 9432, 9510, 9520, 9521, 9530, 9531, 9532, 9540, 9541, 9542, 9543, 9610, 9620, 9621, 9630, 9631, 9632, 9640, 9641, 9642, 9643, 9650, 9651, 9652, 9653, 9654, 9710, 9720, 9721, 9730, 9731, 9732, 9740, 9741, 9742, 9743, 9750, 9751, 9752, 9753, 9754, 9760, 9761, 9762, 9763, 9764, 9765, 9810, 9820, 9821, 9830, 9831, 9832, 9840, 9841, 9842, 9843, 9850, 9851, 9852, 9853, 9854, 9860, 9861, 9862, 9863, 9864, 9865, 9870, 9871, 9872, 9873, 9874, 9875, 9876

Notice how the hexadecimal digits A, B, C, D, E and F are not required to represent the decimal equivalents. As examples we can see that:$$7210_{16} = 29200_{10} \\ 9876_{16}=39030_{10}$$

Trouble With Numbers Aplenty

Unfortunately one of the mathematical sites that I use on a daily basis has been difficult to access recently and today it refuses to open at all. Figure 1 shows the message that appears.

Figure 1

My daily tweet always contains a link to Numbers Aplenty as well as a screenshot. Figure 2 shows yesterday's tweet.

Figure 2

Similarly my AirTable database always contains a link to Numbers Aplenty. I put the link up for today even though it doesn't currently work. See Figure 3.

Figure 3

It would be most unfortunate if the site were to go offline permanently. Let's hope that doesn't happen. I put an update in this post once things are back to normal.

POSTSCRIPT: happily, the site now seems back to normal as very late in the evening of October 1st 2024 (Melbourne time).

Metadromes

To quote from Numbers Aplenty:

A number is a metadrome in a given base \(b\) (often 10 or 16) if its digits are in strictly increasing order in that base. For example, 1234, 68 and 12789 are all metadromes in base 10. The total number of metadromes in base \(b\)  is equal to \(2^{b-1}\), hence in base 10 there are \(2^{10-1} = 2^9=512\) metadromes ranging from 0 to 123456789.  

For some reason, I've ignored these numbers over the years even though they make their appearance regularly in Numbers Aplenty. Here is the full list of base 10 metadromes:

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 56, 57, 58, 59, 67, 68, 69, 78, 79, 89, 123, 124, 125, 126, 127, 128, 129, 134, 135, 136, 137, 138, 139, 145, 146, 147, 148, 149, 156, 157, 158, 159, 167, 168, 169, 178, 179, 189, 234, 235, 236, 237, 238, 239, 245, 246, 247, 248, 249, 256, 257, 258, 259, 267, 268, 269, 278, 279, 289, 345, 346, 347, 348, 349, 356, 357, 358, 359, 367, 368, 369, 378, 379, 389, 456, 457, 458, 459, 467, 468, 469, 478, 479, 489, 567, 568, 569, 578, 579, 589, 678, 679, 689, 789, 1234, 1235, 1236, 1237, 1238, 1239, 1245, 1246, 1247, 1248, 1249, 1256, 1257, 1258, 1259, 1267, 1268, 1269, 1278, 1279, 1289, 1345, 1346, 1347, 1348, 1349, 1356, 1357, 1358, 1359, 1367, 1368, 1369, 1378, 1379, 1389, 1456, 1457, 1458, 1459, 1467, 1468, 1469, 1478, 1479, 1489, 1567, 1568, 1569, 1578, 1579, 1589, 1678, 1679, 1689, 1789, 2345, 2346, 2347, 2348, 2349, 2356, 2357, 2358, 2359, 2367, 2368, 2369, 2378, 2379, 2389, 2456, 2457, 2458, 2459, 2467, 2468, 2469, 2478, 2479, 2489, 2567, 2568, 2569, 2578, 2579, 2589, 2678, 2679, 2689, 2789, 3456, 3457, 3458, 3459, 3467, 3468, 3469, 3478, 3479, 3489, 3567, 3568, 3569, 3578, 3579, 3589, 3678, 3679, 3689, 3789, 4567, 4568, 4569, 4578, 4579, 4589, 4678, 4679, 4689, 4789, 5678, 5679, 5689, 5789, 6789, 12345, 12346, 12347, 12348, 12349, 12356, 12357, 12358, 12359, 12367, 12368, 12369, 12378, 12379, 12389, 12456, 12457, 12458, 12459, 12467, 12468, 12469, 12478, 12479, 12489, 12567, 12568, 12569, 12578, 12579, 12589, 12678, 12679, 12689, 12789, 13456, 13457, 13458, 13459, 13467, 13468, 13469, 13478, 13479, 13489, 13567, 13568, 13569, 13578, 13579, 13589, 13678, 13679, 13689, 13789, 14567, 14568, 14569, 14578, 14579, 14589, 14678, 14679, 14689, 14789, 15678, 15679, 15689, 15789, 16789, 23456, 23457, 23458, 23459, 23467, 23468, 23469, 23478, 23479, 23489, 23567, 23568, 23569, 23578, 23579, 23589, 23678, 23679, 23689, 23789, 24567, 24568, 24569, 24578, 24579, 24589, 24678, 24679, 24689, 24789, 25678, 25679, 25689, 25789, 26789, 34567, 34568, 34569, 34578, 34579, 34589, 34678, 34679, 34689, 34789, 35678, 35679, 35689, 35789, 36789, 45678, 45679, 45689, 45789, 46789, 56789, 123456, 123457, 123458, 123459, 123467, 123468, 123469, 123478, 123479, 123489, 123567, 123568, 123569, 123578, 123579, 123589, 123678, 123679, 123689, 123789, 124567, 124568, 124569, 124578, 124579, 124589, 124678, 124679, 124689, 124789, 125678, 125679, 125689, 125789, 126789, 134567, 134568, 134569, 134578, 134579, 134589, 134678, 134679, 134689, 134789, 135678, 135679, 135689, 135789, 136789, 145678, 145679, 145689, 145789, 146789, 156789, 234567, 234568, 234569, 234578, 234579, 234589, 234678, 234679, 234689, 234789, 235678, 235679, 235689, 235789, 236789, 245678, 245679, 245689, 245789, 246789, 256789, 345678, 345679, 345689, 345789, 346789, 356789, 456789, 1234567, 1234568, 1234569, 1234578, 1234579, 1234589, 1234678, 1234679, 1234689, 1234789, 1235678, 1235679, 1235689, 1235789, 1236789, 1245678, 1245679, 1245689, 1245789, 1246789, 1256789, 1345678, 1345679, 1345689, 1345789, 1346789, 1356789, 1456789, 2345678, 2345679, 2345689, 2345789, 2346789, 2356789, 2456789, 3456789, 12345678, 12345679, 12345689, 12345789, 12346789, 12356789, 12456789, 13456789, 23456789, 123456789 

Of these 512 metadromes, 100 or about 20% are prime. \(p_{13479}=145679\) is the largest metadromic prime whose index is a metadrome too. Here are the primes:

2, 3, 5, 7, 13, 17, 19, 23, 29, 37, 47, 59, 67, 79, 89, 127, 137, 139, 149, 157, 167, 179, 239, 257, 269, 347, 349, 359, 367, 379, 389, 457, 467, 479, 569, 1237, 1249, 1259, 1279, 1289, 1367, 1459, 1489, 1567, 1579, 1789, 2347, 2357, 2389, 2459, 2467, 2579, 2689, 2789, 3457, 3467, 3469, 4567, 4679, 4789, 5689, 12347, 12379, 12457, 12479, 12569, 12589, 12689, 13457, 13469, 13567, 13679, 13789, 15679, 23459, 23567, 23689, 23789, 25679, 34589, 34679, 123457, 123479, 124567, 124679, 125789, 134789, 145679, 234589, 235679, 235789, 245789, 345679, 345689, 1234789, 1235789, 1245689, 1456789, 12356789, 23456789

Of these 512 metadromes, 131 have prime factors that are also metadromes. The largest of these is \(1235689 = 7 \times 13 \times 37 \times 367\). Here are the numbers:

1, 4, 6, 8, 9, 12, 14, 15, 16, 18, 24, 25, 26, 27, 28, 34, 35, 36, 38, 39, 45, 46, 48, 49, 56, 57, 58, 68, 69, 78, 125, 126, 128, 134, 135, 136, 138, 145, 147, 148, 156, 158, 168, 169, 178, 189, 234, 235, 236, 237, 238, 245, 247, 256, 259, 267, 268, 278, 289, 345, 348, 356, 357, 358, 368, 378, 456, 459, 468, 469, 478, 567, 578, 1235, 1239, 1246, 1248, 1256, 1258, 1269, 1345, 1357, 1368, 1369, 1456, 1458, 1468, 1479, 1568, 2345, 2346, 2349, 2368, 2457, 2478, 2479, 2569, 2578, 2679, 3456, 3458, 3468, 3478, 3578, 5678, 12348, 12358, 12467, 12478, 12789, 13456, 13467, 13468, 13579, 13689, 15678, 23569, 24589, 24678, 24679, 25678, 34568, 34569, 124579, 124689, 134568, 134589, 134689, 234567, 234689, 1235689

There are only 30 numbers that are metadromes in base 8 and base 10 and these are (permalink):

1, 2, 3, 4, 5, 6, 7, 12, 13, 14, 15, 19, 23, 28, 29, 37, 38, 39, 46, 47, 156, 157, 158, 159, 167, 238, 239, 247, 678, 679

See Table 1 for the conversions to base 8. There are of course only \(2^{8-1} = 2^7 =128 \) metadromes in base 8. 

Table 1: permalink

The algorithm that the permalink refers to can be used to generate similar tables for bases 9, 7, 6, 5, 4, 3 and 2, although for 2 there is only the number 1 that satisfies. The algorithm will not work for bases greater than 10, although it would be interesting to try to develop one.

Properties of Concatenated Numbers

Let's split a number in two parts and then consider the relationship that these two parts bear to the number of which both are a part. Now I've done this before as described in a post titled Energetic Numbers. In THAT post, I looked at numbers that belong to OEIS  A055480:

A055480

Energetic numbers: numbers that can be broken into two or more substrings and expressed as a sum of (possibly different) positive powers of those substrings.

In THIS post I'll be looking at two different types of relationships. Let's consider 27573 and break it into two parts: 275 and 73. It turns out that my diurnal age today (27573) has an interesting property and it involves the totient function, commonly referred to as Euler's phi function or simply the phi function. We find that:$$ \begin{align}  \phi(27573) &= 1440 \\ \phi(275) \times \phi(73) &= 200 \times 72\\ &=1440 \end{align} $$Numbers like this belong to OEIS A147619 :

 A147619             Numbers \(n\)  = \(a \,| \, b\) such that \( \phi(n) = \phi(a) \times \phi(b) \)
                                where the symbol | represents concatenation

The initial members of this sequence are:

78, 780, 897, 918, 1179, 1365, 1776, 2574, 2598, 2967, 3168, 3762, 4758, 5775, 5796, 7800, 7875, 7917, 8217, 8970, 9180, 9576, 11790, 13650, 13662, 13875, 13896, 14391, 17760, 18564, 18858, 19812, 20097, 25740, 25935, 25974, 25980, 27573, 28776, 28779, 29670, 31680, 33165, 35919, 37620

Let's look at the breakdown for the numbers between 27573 and 37620:

• 275 | 73  -->  27573
  
  
•  287 | 76  -->  28776
  
  
•  287 |  79  -->  28779
  
  
•  29 | 670  -->   29670
  
  
•  31 | 680  -->   31680
  
  
•  331 | 65  -->   33165
  
  
•  35 | 919  -->   35919
  
  
•  37 | 620  -->   37620

A similar thing can be done with the sigma function that tallies the sum of the divisors of a number. Once again we break the number into two parts and then use the sigma function to connect those two parts to the whole. There is no OEIS function this time around but the property of the numbers being considered is as follows:

 Numbers \(n\)  = \(a \,| \, b\) such that \( \sigma(n) = \sigma(a) \times \sigma(b) \)
where the symbol | represent concatenation

The 78 numbers satisfying this condition between 27000 and 40000 are (permalink):

27175, 27298, 27418, 27445, 27500, 28018, 28195, 28750, 28798, 28978, 29038, 29058, 29098, 29278, 29395, 29398, 29500, 29875, 29875, 29950, 29950, 29980, 30498, 30775, 30788, 30989, 31750, 31795, 31918, 32595, 33175, 33238, 33298, 33725, 34189, 34555, 34557, 34795, 34975, 35338, 35395, 35578, 35585, 35670, 35818, 35975, 36178, 36238, 36350, 36398, 36426, 36775, 37138, 37241, 37678, 37750, 37798, 38247, 38370, 38518, 38570, 38638, 38750, 38826, 38856, 38877, 38940, 38975, 39118, 39425, 39478, 39500, 39754, 39805, 39898, 39976, 39980, 39992

Table 1 shows the breakdown for numbers between 27000 and 30000:

Table 1

Let's look at the first number 27175 where we break the number into 271 and 75 and we have:$$ \begin{align} \sigma(27175) &= 33728\\ \sigma(271) \times \sigma(75) &= 272 \times 124\\ &= 33728 \end{align}$$There are other possible concatenations involving more than two parts or other functions and properties of numbers so this post is merely a sample.

Palindromic Day 27572

Every 100 days another palindrome day rolls by and yesterday I celebrated palindromic day 27572. Now this palindrome has an arithmetic digital root that is equal to its central digit of 5. This is because:$$ 27572 \rightarrow 2+7+5+7+2= 23 \rightarrow 2 + 3 =5 $$However, the absolute difference between the first two digits (and of course the last two digits as well) is also equal to the digital root and the central digit.

| 2 - 7 | = 5 = | 7 - 2 |

This makes the palindrome extra special and in the range of five digit numbers from 10000 to 99999 only the following palindromes have the properties previously mentioned. These are (permalink):

18781, 27572, 36363, 45154, 54145, 63336, 72527, 81718, 90909

If we allowed leading zeros then we would have:

• 09990 has the same digits as 90909

| 0 - 9 | = 9 = | 9 - 0 |

• 18781 has the same digits as 81718

| 1 - 8 | = 7 = | 8 - 1 |

• 27572 has the same digits as 72527

| 2 - 7 | = 5 = | 7 - 2 |

• 36363 has the same digits as 63336

| 3 - 6 | = 3 = | 6 - 3 |

• 45154 has the same digits as 54145

| 4 - 5 | = 1 = | 5 - 4 |

Remember that the central digit is also the arithmetic digital root of the number.

Digit Permutations of Numbers

I've written about permutations of the digits of a number in many posts. For example, in my recent post titled Determinants of Circulant Matrices (11th of August 2024) I looked at circulant matrices that have determinants with digits that are permutations of the number's digits. In this current post, I'll be looking at some new types, specifically where the digits of the number are equal to:

• a permutation of the digits of the aliquot sum of the divisors of the number
• a permutation of the digits of the sum of the divisors of the number
• a permutation of the digits of the totient of the number

SUM OF PROPER DIVISORS

There are 37 numbers in the range up to 40,000 that have the same digits as the sum of their proper divisors. These numbers are listed below together with the permutation. Notice that perfect numbers remain unchanged. Permalink.

(6, 6), (28, 28), (411, 141), (496, 496), (604, 460), (1305, 1035), (3664, 3466), (4086, 4806), (4672, 4726), (4896, 9846), (5046, 5406), (7785, 5787), (8128, 8128), (8739, 3897), (9331, 1933), (14535, 13545), (16012, 12016), (18342, 21438), (18585, 18855), (19648, 19468), (20634, 23046), (21534, 23154), (21628, 16228), (22365, 22563), (25911, 11529), (27568, 25876), (28108, 21088), (29160, 69210), (29188, 21898), (31185, 38511), (32091, 13029), (32271, 12273), (34956, 53496), (35898, 38598), (35925, 23595), (36172, 27136), (37698, 39678)

These numbers constitute OEIS A085844 and they are:

6, 28, 411, 496, 604, 1305, 3664, 4086, 4672, 4896, 5046, 7785, 8128, 8739, 9331, 14535, 16012, 18342, 18585, 19648, 20634, 21534, 21628, 22365, 25911, 27568, 28108, 29160, 29188, 31185, 32091, 32271, 34956, 35898, 35925, 36172, 37698

SUM OF DIVISORS

There are 45 numbers in the range up to 40,000 that have the same digits as the sum of their divisors including the number itself this time. These numbers are listed below together with the permutations. Permalink.

(1, 1), (69, 96), (258, 528), (270, 720), (276, 672), (609, 960), (639, 936), (2391, 3192), (2556, 6552), (2931, 3912), (3409, 3904), (3678, 7368), (3679, 3976), (4291, 4912), (5092, 9520), (6937, 7936), (8251, 8512), (10231, 11032), (12087, 18720), (12931, 13192), (15480, 51480), (16387, 18736), (20850, 52080), (22644, 62244), (22893, 32928), (24369, 32496), (26145, 52416), (26442, 62244), (27846, 78624), (28764, 78624), (29880, 98280), (29958, 59928), (30823, 33208), (31812, 81312), (32658, 65328), (34207, 34720), (34758, 75348), (34909, 39904), (36045, 65340), (36249, 49632), (36729, 67392), (36978, 73968), (36990, 99360), (38491, 39184), (38538, 83538)

These numbers constitute OEIS A115920 and they are:

1, 69, 258, 270, 276, 609, 639, 2391, 2556, 2931, 3409, 3678, 3679, 4291, 5092, 6937, 8251, 10231, 12087, 12931, 15480, 16387, 20850, 22644, 22893, 24369, 26145, 26442, 27846, 28764, 29880, 29958, 30823, 31812, 32658, 34207, 34758, 34909, 36045, 36249, 36729, 36978, 36990, 38491, 38538

TOTIENT

There are 36 numbers in the range up to 40,000 that have the same digits as their totients. These numbers are listed below together with the permutations. Permalink.

(1, 1), (21, 12), (63, 36), (291, 192), (502, 250), (2518, 1258), (2817, 1872), (2991, 1992), (4435, 3544), (5229, 2952), (5367, 3576), (5637, 3756), (6102, 2016), (6174, 1764), (6543, 4356), (6822, 2268), (7236, 2376), (7422, 2472), (8022, 2280), (8541, 5184), (8982, 2988), (17631, 11736), (18231, 11832), (18261, 12168), (20301, 13200), (20518, 10258), (20617, 20176), (21058, 10528), (22471, 21472), (22851, 15228), (25196, 12596), (25918, 12958), (27615, 12576), (29817, 19872), (34816, 16384), (35683, 33568)

These numbers constitute OEIS A115921 and all except 1 are deficient. They are:

1, 21, 63, 291, 502, 2518, 2817, 2991, 4435, 5229, 5367, 5637, 6102, 6174, 6543, 6822, 7236, 7422, 8022, 8541, 8982, 17631, 18231, 18261, 20301, 20518, 20617, 21058, 22471, 22851, 25196, 25918, 27615, 29817, 34816, 35683