Daniel´s number!
Daniel´s number (DN42) is a large number, defined by the Brazilian "amateur" googologist Daniel Corrêa.
The Daniel´s number construction starts with 42, being inspired by the Hitchhiker's Guide to Galaxy science fiction series, where a supercomputer named Deep Thought, after spends 7.5 million years computing the Answer to the Ultimate Question of Life, the Universe, and Everything, arriving at the answer 42.
1-) Let 00D = 42;
2-) Let 01D = 42↑↑...↑↑42, where the number of ↑´s is 42;
3-) Let 02D = (01D)↑↑...↑↑(01D), where the number of ↑´s is 01D;
4-) Let 0(k+1)D = (0kD)↑↑...↑↑(0kD), where the number of ↑´s is 0kD, continue until k = 41, resulting in 10D;
5-) Let 11D = (10D)↑↑...↑↑(10D), where the number of ↑´s is 10D;
6-) Let 1(k+1)D = (1kD)↑↑..…
Steinhaus-Moser: some 2's in step-by-step.
Steinhaus-Moser Notation is a well know notation created by Hugo Steinhaus, and extended by Leo Moser.
My intention in this post is to show the resolution of some 2's inside polygons of Steinhaus-Moser notation. It´s well know that the first polygon is the triangle, and any number \(n\) inside a triangle is converted to the \(n\) power of \(n\) or \(n^{n}\).
Here we represent a value \(n\) inside a triangle as \(n[3]_{1}\), according the general notation proposed by Susan Stepney. The notation extends to:
- \(n\) inside a square or \(n[4]_{1}\) = \(n\) inside \(n\) triangles = \(n[3]_{n}\);
- \(n\) inside a pentagon or \(n[5]_{1}\) = \(n\) inside \(n\) squares = \(n[4]_{n}\);
- \(n\) inside a hexagon or \(n[6]_{1}\) = \(n\) inside \(n\) pentagons = \…
Sumselfgogopyr
Sumselfgogopyr is defined as the summation of the n-th n-gonal pyramidal numbers with n from 3 to gogopyr, the term was coined by Daniel Corrêa.
Sumselfgogopyr is computed as described below:
\(Sumselfgogopyr = \sum^{gogopyr}_{n=3} \frac{3 \cdot n^{2} + n^{3} \cdot (n-2) - n \cdot (n-5)}{6}\)
Sumselfgogopyr has 1995 digits, and according to the calculations using BCalc to convert gogopyr to power of 10 notation, and using WolframAlpha to perform the calculation with the maximum input length, the approximated form of sumselfgogopyr is:
\(Sumselfgogopyr \approx 4.28669410150... \times 10^{1994}\)
Selfgogopyr
Selfgogopyr is defined as the gogopyrth gogopyrgonal pyramidal number. The term was coined by Daniel Corrêa.
Selfgogopyr is computed using the general formula for an r-gonal pyramidal number described below:
\([3.n^{2} + n^{3}.(r-2) - n.(r-5)]/6\)
where \(r = n = gogopyr\).
Selfgogopyr has 1597 digits, and according to the calculations performed using BCalc configured to show 2011 digits, the full decimal form is:
Sumgogopyr
Sumgogopyr is defined as the summation of the n-th n-gonal pyramidal numbers with n from 3 to googol, the term was coined by Daniel Corrêa.
Sumgogopyr is computed as described below:
\(Sumgogopyr = \sum^{googol}_{n=3} \frac{3 \cdot n^{2} + n^{3} \cdot (n-2) - n \cdot (n-5)}{6}\)
In its decimal form, sumgogopyr has 499 digits as described: 3 as the first 299 digits followed by the digit 8 followed by a sequence of 99 3's followed by the digit 7 followed by a sequence of 98 9's, and 6 as the last digit.
The full decimal form of sumgogopyr is expressed below: