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:
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:
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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.
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