One Step Away

I've written about home primes before in the following posts:

• Concatenations on June 11th 2021
• Home Primes on May 2nd 2021
• Prime Weeks on March 8th 2020 (refurbished on May 2nd 2021)

I was reminded of the term again today when I was searching for properties associated with the number representing my diurnal age: 26601. This was one of those numbers that crop up from time to time that seemingly, after exhausting all my usual resources, have no interesting properties. Of course, they invariably do and it's up to me to discover them.

After some time, I hit on the idea of home primes. These are the primes formed when the factors of a composite number are concatenated to form a new number. The process is continued until a prime number is reached. For example, let's consider the number 9 that has a home prime of 311.

9 = 3 x 3 → 33 = 3 x 11 → 311 which is prime

It occurred to me that 26601 = 3 x 8867 --> 38867 is prime. In other words, it only takes one step for 26601 to get to its home prime. Some numbers, the smallest being 49, haven't been shown as yet to reach a home prime but I've written about these Home Primes post. What I was interested in here was finding all the numbers in the range up to 40,000 that reach their home primes in only one step.

I developed a SageMath algorithm (permalink) to find these numbers and it turns out that there are 6699 of them, representing about 16.7% of the total numbers in the range. These numbers form OEIS A046411: 

A046411

Composite numbers the concatenation of whose prime factors is a prime.

The initial members of the sequence are:

6, 12, 18, 21, 22, 28, 33, 39, 46, 51, 52, 54, 58, 63, 66, 70, 82, 84, 93, 98, 111, 115, 117, 133, 141, 142, 148, 154, 159, 162, 165, 166, 171, 172, 175, 177, 182, 187, 198, 201, 205, 207, 210, 219, 220, 226, 232, 235, 237, 245, 246, 247, 249, 253, 255, 261, 262, 264, 266, 267, 268, 274, 279, 282, 291, 292, 294, 297, 301, 310, 319, 327, 338, 350, 355, 358, 376, 384, 385, 387, 388, 391, 392, 399, 406, 408, 411, 423, 426, 427, 430, 432, 434, 435, 436, 440, 442, 459, 468, 472, 475, 476, 478, 489, 494, 498, 501, 502, 504, 505, 511, 516, 525, 531, 534, 535, 538, 543, 549, 552, 562, 565, 568, 573, 574, 583, 584, 586, 589, 595, 598, 608, 615, 620, 622, 628, 630, 632, 639, 657, 664, 679, 684, 686, 687, 694, 696, 697, 705, 721, 728, 741, 742, 752, 753, 756, 759, 763, 766, 771, 772, 775, 778, 781, 783, 786, 790, 793, 798, 799, 801, 804, 813, 816, 819, 820, 833, 835, 837, 856, 860, 871, 872, 875, 884, 885, 888, 889, 895, 901, 904, 916, 921, 924, 930, 934, 938, 939, 943, 946, 949, 963, 968, 985, 993, ...

So I had my "interesting" property now for 26601 and duly tweeted about it. See Figure 1:

Figure 1: source

I was interested as expected in runs of such numbers. How many pairs, triplets, quadruplets etc. are there in a given range. I decided to extend my range to one million and see what came up. What I found is that there are:

• 137125 singletons representing 13.7% of the numbers in the range
  (remember it was 16.7% in the range up to 40,000): link
  
  
• 18854 pairs of numbers representing 37708 numbers or 3.77%: link
  
  
• 2557 triplets representing 7671 numbers or 0.767%: link
  
  
• 358 quadruplets representing 1432 numbers or 0.143%
  
  
• 50 quintuplets representing 250 numbers or 0.0250%
  
  
• 9 sextuplets representing 54 numbers or 0.00540%
  
  
• 2 septuplets representing 14 numbers or 0.00140%
  
  
• 1 octuplet representing 8 numbers or 0.000800%

These pairs, triplets, quadruplets, quintuplets, sextuplets, septuplets and octuplets are not necessarily stand-alone. For example, the two septuplets, beginning with 45162 and 45163, together form the octuplet 45162, 45163, 45164, 45165, 45166, 45167, 45168, 45169 (see post titled What's Special About 45162? uploaded on September 3rd 2022). I could modify the algorithm to generate stand-alone groups but I think the overview I've given is sufficient.

Concatenations

Today I turned 26366 days old and yesterday, of course, I was 26365 days old. Each of these numbers have the property that the leading two digits (26) represent the number of fortnights in a year while the final three digits (365 and 366) represents the number of days in non-leap years and a leap years respectively. Incidentally, 26 fortnights represent 26 x 14 = 364 days and so 26364 encodes that fact. Similar pairs of numbers include:

• 12365 and 12366 where 12 represents the number of months in a calendar year
• 13365 and 13366 where 13 represents the number of lunar months in a calendar year
• 52365 and 52366 where 52 represents the number of full weeks in a calendar year

The concatenation of numbers, according to WolframMathWorld, is represented by the symbol \( \parallel \) and thus 26\( \parallel \)366 = 26366. The formula for the concatenation in base \(b\) of two numbers, \(p\) and \(q\), is given by:$$p \parallel q=p \, b^{f(q)} + q \text{ where } f(q)=\left \lfloor \log_b{q} \right \rfloor + 1$$$$ \text{where }f(q) \text{ represents the length of } q$$Let's test this formula out in the case of 26366. We have \(p=26\) and \(q=366\). Clearly the length of \(q\) is 3 but let's check using \( \left \lfloor \log_{10}366 \right \rfloor + 1\). The value of this is indeed 3 so all is good.

When constructing an algorithm, the formula is useful as a way of accomplishing the concatenation without resorting to strings. Figure 1 shows the relevant SageMath code.

Figure 1: permalink

26365 and 26366 have some interesting shared properties. They are both semiprimes and emirpimes :

• 26365 = 5 * 5273 and 56362 = 2 * 28181
• 26366 = 2 * 13183 and 66362 = 2 * 33181

In recreational mathematics, there are some interesting applications of concatenation. One of these involves so-called home primes. These are primes obtained by repeatedly factoring the increasing concatenation of prime factors of a given number. I've written about these in an eponymous post on May 2nd 2021. Let's work out the home primes for 26364, 26365 and 26366.

For 26364, we find ten steps are required to reach the home prime: 

 

26364

223131313

792824447

10113956473

21147933443

713589739409

4059117579999

31353039193333

1113305413717887

313175353597790561

For 26365, only six steps are required:

26365

55273

311783

734271

3311787

31764937

For 26366, seven steps are required:

26366

213183

3323687

17195511

35731837

196728069

365576023

The Smarandache–Wellin numbers involve the concatenations of the first prime numbers. These numbers form OEIS A019518:

A019518

Smarandache-Wellin numbers: a(n) is the concatenation of first n primes (written in base 10).

The sequence begins:

2

23

235

2357

235711

23571113

2357111317

235711131719

23571113171923

2357111317192329

235711131719232931

23571113171923293137

2357111317192329313741

235711131719232931374143

23571113171923293137414347

Of the numbers listed here only 2, 23 and 2357 are prime. The next such number that is prime has 355 digits!

If we concatenate the integers, we create the Champernowne constant, a transcendental real constant whose decimal expansion has important properties. It is named after economist and mathematician D. G. Champernowne, who published it as an undergraduate in 1933. I've written about this constant in an eponymous post on March 22nd 2019. The constant begins 12345678910111213141516 ... 

The Copeland–Erdős constant is formed by the concatenation of "0." with the base 10 representations of the prime numbers in order. Its value, using the modern definition of prime, is approximately 0.235711131719232931374143…

There is a reverse integer sequence that comprises OEIS A000422:

A000422

Concatenation of numbers from n down to 1.

The sequence begins:

1

21

321

4321

54321

654321

7654321

87654321

987654321

10987654321

1110987654321

121110987654321

13121110987654321

1413121110987654321

151413121110987654321

16151413121110987654321

1716151413121110987654321

181716151413121110987654321

Interestingly, this sequence produces very few primes with the first being when \(n=82\) and the next being \(n=37765\).

We can concatenate the odd, even, triangular, square, cubic and Fibonacci numbers and all of these have associated OEIS sequences. Follow this link for more information.

Home Primes

I first mentioned Home Primes in a post from Sunday, March 8th 2020, titled Prime Weeks. As the Wikipedia entry states regarding this class of primes:

Investigations into home primes make up a minor side issue in number theory. Its questions have served as test fields for the implementation of efficient algorithms for factoring composite numbers, but the subject is really one in recreational mathematics.

So let's put serious mathematics aside and delve into this corner of recreational mathematics. Let's begin with the definition, from the same source, that runs:

In number theory, the home prime HP(\(n\)) of an integer \(n\) greater than 1 is the prime number obtained by repeatedly factoring the increasing concatenation of prime factors including repetitions. 

Using SageMath, I was able to create an algorithm to determine the home prime for any composite integer. See Figure 1 with permalink attached.

Figure 1: permalink

As can be seen, the number 26327 (my diurnal age on the day of this post's creation) leads, after eight steps, to its home prime of 3325112912503. Apart from the fact that it represents my diurnal age, the reason that I chose 26327 as my example above is that it factorises to 7 x 3761 and the latter number has an important connection to home primes because it is a member of OEIS A118756:

A118756

a(\(n\)) = smallest prime \(p\) such that \(p\) is the home prime of exactly \(n\) natural numbers.

The sequence begins: 2, 23, 211, 379, 773, 3389, 23251, 3761, ... and so 3761 is in the position corresponding to \(n=8\). This means that there are seven composite numbers whose home prime is 3761. These are listed in the OEIS comments section for the sequence. The eight numbers are:

140, 332, 514, 566, 1281, 2257, 2283, 3761

Let's now consider OEIS A037274 that lists the home primes (if known) for all the natural numbers:

A037274

Home primes: for n >= 2, a(n) = the prime that is finally reached when you start with n, concatenate its prime factors (A037276) and repeat until a prime is reached (a(n) = -1 if no prime is ever reached).

Here is the sequence up to \(n=48\):

1, 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, 11, 223, 13, 13367, 1129, 31636373, 17, 233, 19, 3318308475676071413, 37, 211, 23, 331319, 773, 3251, 13367, 227, 29, 547, 31, 241271, 311, 31397, 1129, 71129, 37, 373, 313, 3314192745739, 41, 379, 43, 22815088913, 3411949, 223, 47, 6161791591356884791277

Looking at the terms in this sequence, it can be seen that the home prime for \(n=8\) is a staggering 3331113965338635107. The progression is as follows:

8 -> 2 * 2 * 2 

222 -> 2 * 3 * 37 

2337 -> 3 * 19 * 41 

31941 -> 3 * 3 * 3 * 7 * 13 * 13 

33371313 -> 3 * 11123771 

311123771 -> 7 * 149 * 317 * 941 

7149317941 -> 229 * 31219729 

22931219729 -> 11 * 2084656339 

112084656339 -> 3 * 347 * 911 * 118189 

3347911118189 -> 11 * 613 * 496501723 

11613496501723 -> 97 * 130517 * 917327 

97130517917327 -> 53 * 1832651281459 

531832651281459 -> 3 * 3 * 3 * 11 * 139 * 653 * 3863 * 5107

3331113965338635107 is prime, so a(8) = 3331113965338635107

However, upon reaching \(n=49\), we encounter the first number that has no known home prime. The composite numbers deriving from 49 and that so far do not lead to a prime are listed in OEIS A056938:

A056938

Concatenate all the prime divisors in previous term (with repetition), starting at 49.

The initial terms are:

49, 77, 711, 3379, 31109, 132393, 344131, 1731653, 71143523, 11115771019, 31135742029, 717261644891, 11193431873899, 116134799345907, 3204751189066719, 31068250396355573, 62161149980213343, 336906794442245927, 734615161567701999, 31318836286194043641 

IMPORTANT RESOURCES

• https://oeis.org/A037274/a037274_1.txt 

This site provides a list of numbers up to 10000 together with the progressions to their respective home primes (if known). 

• https://www.rieselprime.de/Others/HomePrime10.htm

This site lists the 471 numbers less than 10000 for which there are no known home primes. In TABLE 1 below I've copied and pasted the first column. The numbers with backgrounds shaded in green may have home primes but not those unshaded (as far as is known). For example, 669 leads to a home prime after 46 steps. I developed a modified algorithm to take into account numbers that don't end in a prime after a certain number of cycles (50 seems to be about the limit for SageMathCell). See Figure 2 that uses 669 as an example. Permalink attached.

Figure 2: permalink

TABLE 1

49
112
146
234
242
284
300
312
320
322
326
328
336
352
363
372
412
460
495
548
556
576
596
663
665
669
670
693
712
714
715
744
749
762
768
782
796
845
847
858
861
867
896
925
973
978
984
992
1008
1030
1053
1067
1138
1139
1220
1248
1298
1314
1315
1316
1328
1370
1394
1408
1416
1444
1448
1452
1455
1456
1467
1515
1519
1521
1529
1539
1552
1565
1568
1595
1596
1610
1628
1672
1681
1726
1734
1751
1757
1772
1781
1782
1846
1855
1897
1908
1915
1956
1964
1980
1985
2008
2021
2025
2040
2048
2068
2071
2104
2105
2112
2114
2117
2138
2147
2148
2172
2189
2206
2248
2252
2262
2265
2280
2316
2320
2321
2360
2390
2410
2432
2436
2442
2465
2480
2484
2510
2520
2558
2560
2581
2594
2611
2618
2637
2642
2658
2660
2662
2684
2686
2745
2774
2784
2796
2800
2802
2812
2816
2842
2848
2888
2922
2965
2993
3006
3016
3024
3036
3038
3056
3068
3102
3108
3110
3136
3158
3168
3172
3192
3200
3208
3210
3215
3238
3250
3252
3262
3270
3278
3285
3286
3288
3296
3304
3332
3336
3363
3368
3370
3386
3388
3393
3408
3429
3451
3454
3466
3470
3475
3489
3492
3495
3498
3520
3586
3604
3606
3662
3675
3685
3720
3721
3755
3766
3776
3782
3790
3800
3806
3813
3816
3835
3836
3852
3858
3867
3868
3879
3894
3905
3909
3916
3955
3966
3968
3970
4012
4020
4036
4046
4048
4050
4060
4065
4067
4097
4108
4109
4122
4145
4172
4186
4188
4203
4205
4224
4225
4230
4240
4257
4260
4300
4301
4318
4320
4352
4436
4440
4442
4454
4470
4480
4494
4497
4500
4532
4541
4556
4557
4559
4574
4580
4581
4582
4632
4687
4695
4719
4746
4776
4777
4790
4791
4836
4852
4883
4884
4887
4890
4891
4911
4927
4930
4936
4941
4944
4946
4980
4986
4992
5010
5029
5037
5055
5060
5090
5110
5115
5116
5145
5152
5154
5160
5180
5182
5185
5187
5194
5214
5256
5278
5280
5284
5301
5312
5320
5324
5368
5370
5377
5400
5403
5406
5416
5432
5460
5465
5488
5494
5496
5536
5551
5564
5608
5620
5621
5632
5652
5668
5670
5674
5680
5692
5698
5708
5720
5736
5740
5756
5760
5778
5792
5800
5845
5856
5858
5866
5871
5874
5882
5905
5910
5912
5916
5931
5949
6013
6024
6063
6069
6077
6078
6084
6094
6100
6102
6138
6146
6149
6154
6175
6176
6205
6210
6212
6223
6227
6244
6256
6294
6314
6328
6332
6333
6336
6351
6366
6384
6386
6400
6405
6409
6414
6436
6444
6445
6448
6452
6454
6467
6495
6516
6533
6541
6548
6550
6561
6572
6612
6615
6666
6670
6671
6680
6710
6729
6785
6811
6812
6816
6819
6820
6836
6840
6859
6864
6873
6877
6906
6919
6932
6944
6972
6985
6989
7009
7017
7024
7032
7048
7049
7056
7059
7065
7072
7089
7091
7092
7096
7119
7128
7168
7188
7194
7203
7204
7210
7248
7267
7282
7316
7343
7345
7360
7385
7394
7395
7403
7465
7494
7506
7540
7554
7562
7615
7628
7648
7654
7668
7686
7696
7707
7714
7720
7749
7765
7770
7776
7824
7826
7832
7864
7865
7872
7880
7896
7902
7904
7923
7965
7972
7987
8000
8008
8020
8049
8085
8112
8114
8120
8127
8134
8155
8170
8178
8189
8193
8222
8250
8255
8262
8277
8324
8349
8375
8382
8393
8398
8448
8449
8457
8475
8506
8516
8540
8546
8558
8560
8562
8569
8575
8585
8592
8603
8608
8610
8618
8624
8635
8668
8670
8680
8696
8704
8709
8710
8729
8734
8736
8739
8740
8757
8763
8784
8790
8793
8806
8824
8846
8873
8874
8890
8909
8932
8934
8940
8943
8949
8960
9002
9012
9025
9027
9048
9075
9080
9086
9088
9095
9104
9119
9120
9130
9141
9145
9158
9189
9262
9266
9273
9282
9295
9316
9328
9339
9350
9373
9412
9446
9448
9449
9458
9482
9486
9487
9495
9499
9502
9513
9524
9529
9570
9577
9585
9588
9591
9594
9595
9600
9603
9620
9624
9630
9644
9648
9651
9659
9667
9685
9688
9690
9702
9724
9732
9740
9747
9761
9773
9780
9792
9812
9815
9836
9863
9874
9877
9881
9890
9900
9922
9937
9951
9961
9964
9965
9975
9978
9983
9998
10014
10018
10024
10031
10045
10048
10050
10052
10058
10064
10081
10090
10094
10098
10132
10136
10158
10179
10200
10203
10208
10218
10234
10240
10244
10266
10270
10272
10302
10304
10305
10312
10332
10335
10353
10374
10380
10395
10409
10411
10415
10424
10437
10467
10478
10485
10490
10492
10496
10523
10530
10538
10541
10542
10566
10572
10577
10579
10580
10595
10609
10610
10616
10620
10624
10640
10650
10659
10668
10676
10683
10700
10703
10748
10755
10764
10773
10825
10829
10830
10832
10836
10845
10850
10863
10864
10865
10872
10895
10913
10914
10948
10956
10969
10971
10972
10976
10982
10991
11000
11011
11024
11040
11067
11080
11130
11133
11136
11168
11183
11187
11189
11244
11247
11254
11256
11275
11280
11286
11293
11334
11370
11378
11440
11476
11496
11500