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Figure 1 |
- a digit divided into the next digit e.g. 2 | 4 = 2 which reads 2 divided into 4
- a digit divided by the next digit e.g. 4 / 2 = 2 which reads 4 divided by 2
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Figure 1 |
Continuing with Achmad Damar's "104 Number Theory", I found this little problem interesting. The author asks: let
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Figure 1 |
This got me thinking about how this result was obtained. There are 114 odd integers between 3 and 231 inclusive and 7,032,112,662,630 ways to sample 9 reciprocals at a time (that's over seven trillion ways). Nonetheless, when running this program in SageMathCell, the above combination of fractions was quickly spat out and after that the program timed out. Running the same program in my jupyter notebook (to avert the program timing out), no further combinations were generated. Is this combination of reciprocals unique? I'm not sure.
Reading through Achmad Damar's "104 Number Theory", there are two simple number properties that are mentioned early on that are easy to prove but the proofs are elegant to my eye at least. Let's look at them.
PROPERTY ONE
Let
Let's suppose that the statement is true and that:
PROPERTY TWO
Let
Again, let's suppose that the statement is true and that:
I was surprised today to stumble upon a new category of number called energetic numbers. Such number are reasonably common: the 10,000th such number is 103,718. These sorts of numbers form 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. |
It's not obvious from the description in the OEIS that the numbers forming the sequence can in fact be with associated with Pythagorean triads and right angled triangles. Today I turned 27475 days old and one of the properties of this number is its membership of OEIS A334542.
A334542 | Numbers |
27172 with digital root of 1 27272 with digital root of 2 27372 with digital root of 3 27472 with digital root of 4 27572 with digital root of 5 27672 with digital root of 6 27772 with digital root of 7
27872 with digital root of 8
27972 with digital root of 9
I'll soon be 27472 days old and I've gotten into the habit of creating a post for each palindromic day. So what other special properties does this palindrome have?
27472, 13736, 6868, 3434, 1717, 5152, 2576, 1288, 644, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1There are 108 steps required to reach 1. Figure 1 shows a plot of the numbers using a logarithmic scale for the vertical axis.
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Figure 1 |
27472, 29444, 25240, 31640, 50440, 73040, 114448, 117680, 156112, 174224, 163366, 121862, 81418, 40712, 46648, 61352, 53698, 26852, 28210, 36302, 25954, 15086, 8794, 4400, 7132, 5356, 4836, 7708, 6404, 4810, 4766, 2386, 1196, 1156, 993, 335, 73, 1, 0
Figure 2 shows a plot of these numbers.
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Figure 2 |
3, 5, 7, 9, 11, 15, 27, 32, 33, 37, 45, 47, 55, 99, 111, 135, 165, 167, 185, 297, 329, 333, 407, 495, 544, 555, 999, 1169, 1221, 1485, 1665, 2035, 3232, 3663, 4995, 6105, 7849, 10989, 18315
27472, 54945, 50985, 92961, 86922, 75933, 63954, 61974, 82962, 75933
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Figure 4 |
Consider the following:
Ignoring the reversal itself then, up to one million, there is only one other number with these properties and that is 238656:
21168, 23424, 23616, 27456, 41184, 212256, 213192, 215232, 219072, 230208, 236925, 236928, 238656, 251505, 251748, 253824, 255024, 257856, 259968, 271728, 276696, 276768, 291168, 293328, 299808, 373464, 403056, 403488, 404064, 422208, 424116, 424764, 428928, 441045, 441288, 462384, 472608, 492048, 606096, 610688, 612576, 635328, 804168
These numbers form OEIS A109029 (permalink):
A109029 | Numbers that have exactly nine prime factors counted with multiplicity (A046312) whose digit reversal is different and also has 9 prime factors (with multiplicity). |
Consider the Diophantine equation:
Are there other numbers? Well somebody named Wroblewski found 51361 to be a solution when it written as 48150 - 31764 + 27385 + 7590. Thus:
Any multiple of a member is also a member. A member that is not a multiple of another member is called primitive. Using elliptic curves, Jacobi and Madden prove that there are infinitely many primitive members. According to them, the only primitive members less than 222,000 are 5491 (due to Brudno) and 51361 (due to Wroblewski).
The comments then list a number greater than 220,000 and that number is 1347505009. The reason I came across these numbers is that today my diurnal age is 27455 and that is the fifth multiple of 5491.
Today I came across an interesting website via a link in an OEIS entry for 27454, the number associated with my diurnal age as of today's date. Figure 1 shows a screenshot.
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Figure 1 |
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Figure 3 |
E-mail: pdg@worldofnumbers.comWeb Page: http://www.worldofnumbers.com/index.htmlMy name is Patrick De Geest, born on the 9th of October 1956, in Wezembeek-Oppem, Belgium (about 10 km east of Brussels), unmarried, mildly myopic, graduated in architecture but never practiced the profession. Currently I'm an employee working in the aircargo export sector (National Airport Zaventem). I didn't lose my interest in beautiful patterns and proportions though, and managed to transfer it to the field of numbers.Also, through the years, I gradually became familiar with the use of personal computers (no, I'll never sell my first Sinclair ZX81) and learned for programming techniques (basic, assembly, ...). All these 'creativities' culminated recently in a website about recreational mathematics with 'palindromes' as the main topic. I opted for palindromes not because of my length (181 cm), my average weight (77 kg) or my housenumber (141) but because I was attracted by their overall symmetry and the fact that it was a novel and thus insufficiently studied subject. Thanks to many contributors from all over the world the site is still expanding.For the rest I'm a rather quiet individual who likes to read an occasional book, watch a movie, listen to classical music, travel once or twice a year to a near/far exotic destination and bike from time to time when the weather permits.
Anyway the point is that the site contains a wealth of information about curious number properties with Patrick giving the following overview of the site's contents:
In this well-filled website you'll find a multitude of facts and figures about topics from the World!Of Numbers . Don't look for a logical order. It is an amalgamation of randomly gathered numbers, curios, puzzles, palindromes, primes, gems, your much valued contributions and more general information. Enjoy! Patrick De Geest
Like Taneja's papers described in my previous post, there is great content here for future posts to this blog. Getting back to the original OEIS sequence, we see that:
The first nine digits of the decimal part do indeed contain all the digits from 1 to 9. Interestingly I can find no reference to these sorts of calculations of page 7 of "Nine Digits" topic. Perhaps it's on one of the other pages. Numbers like 27454 are part of OEIS A034279:
A034279 | Decimal part of |