## Seventeen

Today it is the first day of July in the year 2017. On this day there is a point in time which can be represented as 1.7.2017, 17:17:17.
To celebrate this symbolically speaking 17-heavy day, I created a list of 17 integer sequences which all contain the number 17.
All sequences were generated using a Python program; the source code can be viewed below or downloaded. Because the following list is formatted using LaTex, the program’s plaintext output can also be downloaded.

1. Prime numbers $n$.
$\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, \dots\}$
2. Odd positive integers $n$ whose number of goldbach sums (all possible sums of two primes) of $n+1$ and $n-1$ are equal to one another.
$\{5, 7, 15, 17, 19, 23, 25, 35, 75, 117, 177, 207, 225, 237, 321, 393, 453, 495, 555, 567, \dots\}$
3. Positive integers $n$ who are part of a Pythagorean triple excluding $0$: $n^2=a^2+b^2$ with integers $a,b>0$.
$\{5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 50, 51, 52, \dots\}$
4. Positive integers $n$ where $\lfloor (n!)^{\frac{1}{n}} \rfloor$ is prime
$\{4, 5, 6, 7, 8, 12, 13, 17, 18, 19, 28, 29, 33, 34, 35, 44, 45, 46, 49, 50, \dots\}$
5. Positive integers $n$ with distance $1$ to a perfect square.
$\{1, 2, 3, 5, 8, 10, 15, 17, 24, 26, 35, 37, 48, 50, 63, 65, 80, 82, 99, 101, \dots\}$
6. Positive integers $n$ where the number of perfect squares including $0$ less than $n$ is prime.
$\{2, 3, 4, 5, 6, 7, 8, 9, 17, 18, 19, 20, 21, 22, 23, 24, 25, 37, 38, 39, \dots\}$
7. Prime numbers $n$ where either $n-2$ or $n+2$ (exclusive) are prime.
$\{3, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, 137, 139, \dots\}$
8. Positive integers $n$ whose three-dimensional vector’s $(n, n, n)$ floored length is prime, $\lfloor \sqrt{3 \cdot n^2} \rfloor$ is prime.
$\{2, 3, 8, 10, 11, 17, 18, 24, 25, 31, 39, 41, 46, 48, 60, 62, 63, 76, 91, 100, \dots\}$
9. Positive integers $n$ who are the sum of a perfect square and a perfect cube (excluding $0$).
$\{2, 5, 9, 10, 12, 17, 24, 26, 28, 31, 33, 36, 37, 43, 44, 50, 52, 57, 63, 65, \dots\}$
10. Positive integers $n$ whose decimal digit sum is the cube of a prime.
$\{8, 17, 26, 35, 44, 53, 62, 71, 80, 107, 116, 125, 134, 143, 152, 161, 170, 206, 215, 224, \dots\}$
11. Positive integers $n$ for which $\text{decimal\_digitsum}(n)+n$ is a perfect square.
$\{2, 8, 17, 27, 38, 72, 86, 135, 161, 179, 216, 245, 275, 315, 347, 432, 467, 521, 558, 614, \dots\}$
12. Prime numbers $n$ for which $\text{decimal\_digitsum}(n^4)$ is prime.
$\{2, 5, 7, 17, 23, 41, 47, 53, 67, 73, 97, 103, 113, 151, 157, 163, 173, 179, 197, 199, \dots\}$
13. Positive integers $n$ where $decimal_digitsum(2 \cdot n)$ is a substring of $n$.
$\{9, 17, 25, 52, 58, 66, 71, 85, 90, 104, 107, 115, 118, 123, 137, 142, 151, 156, 165, 170, \dots\}$
14. Positive integers $n$ whose decimal reverse is prime.
$\{2, 3, 5, 7, 11, 13, 14, 16, 17, 20, 30, 31, 32, 34, 35, 37, 38, 50, 70, 71, \dots\}$
15. Positive integers $n$ who are a decimal substring of $n^n$.
$\{1, 5, 6, 9, 10, 11, 16, 17, 19, 21, 24, 25, 28, 31, 32, 33, 35, 36, 37, 39, \dots\}$
16. Positive integers $n$ whose binary expansion has a prime number of $1$‘s.
$\{3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, \dots\}$
17. Positive integers $n$ whose 7-segment representation uses a prime number of segments.
$\{1, 2, 3, 5, 7, 8, 12, 13, 15, 17, 20, 21, 26, 29, 30, 31, 36, 39, 47, 48, \dots\}$

# Python 2.7.7 Code
# Jonathan Frech, 29th, 30th  of June 2017