## A285494

The On-Line Encyclopedia of Integer Sequences gets regularly updated with new integer sequences. One of the recent updates was contributed by me, A285494.

A285494 is the list of all numbers $k$ so that its digit sum equals its number of distinct prime factors.
A number’s digit sum is the sum of all of its decimal digits. The number $62831853$, for example, has a digit sum of $6+2+8+3+1+8+5+3 = 36$.
A number’s number of distinct prime factors is the number of different prime numbers that multiply together to result in the original number. As an example, $62831853 = 3^2 \cdot 7 \cdot 127 \cdot 7853$, so it has five prime factors of which four are distinct.
Thereby one can conclude that $62831853$ is not an entry in this sequence, as $36 \neq 4$.

The sequence is certainly infinite, as the number $k = 2 \cdot 10^n$ with $n \in \mathbb{N}^*$ has a digit sum of $2 + (0 \cdot n) = 2$ and — because $k = 2^{n+1} \cdot 5^n$ — exactly two distinct prime factors.

In the encyclopedia entry, I provided a Mathematica one-liner to compute the first few entries of this sequence. Since then, I have also written a Python two-liner to achieve the same goal.

(* Mathematica *)
Select[Range[2,10000],Total[IntegerDigits[#]]==Length[FactorInteger[#]]&]
Out = {20, 30, 102, 120, 200, 300, 1002, 1200, 2000, 2001, 2002, 3000, 3010}

# Python 2.7
>>> def p(n):exec"i,f=2,set()\nwhile n>1:\n\tif n%i==0:f.add(i);n/=i;i=1\n\ti+=1";return len(f)
>>> print filter(lambda n:p(n)==sum(map(int,str(n))),range(2,10001))
[20, 30, 102, 120, 200, 300, 1002, 1200, 2000, 2001, 2002, 3000, 3010]


## TI-99/4A Primes

Being a fan of old hardware, I used the TI-99/4A (released in 1981) to calculate some primes.
The code is written in BASIC, the programming language found on most computers of this era.
Further information on the TI can be found in this Wikipedia article.

1   REM TI-99/4A BASIC Code
2   REM Jonathan Frech 20th of May, 2016

## Prime-Generating Formula

(April Fools’!) I came up with this interesting prime-generating formula. It uses the constant $\xi$ and generates the primes in order!

The constant’s approximation.
$\xi = 1.603502629914017832315523632362646507807932231768273436867961017532625344 \dots$

The formula $p_n$ calculates the n-th prime.
$p_n = \lfloor {10^{2 \cdot n} \cdot \sqrt{\xi^3}} \rfloor - \lfloor {10^{2 \cdot (n - 1)} \cdot \sqrt{\xi^3}} \rfloor \cdot 10^2$

The first few values for $p_n$ when starting with $n=0$ are as follows.
$p_{0 \text{ to } 7} = \{2, 3, 5, 7, 11, 13, 17, 19, \dots \}$