# Seventeen

2017-07-01, post № 174

mathematics, #17, #2017, #integer, #integer sequences, #July, #sequences

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. 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.

3. Positive integers n who are part of a Pythagorean triple excluding 𝟢: $n^2=a^2+b^2$ with integers $a,b>0$.

4. Positive integers 𝑛 where $\lfloor(n!)^{\frac{1}{n}}\rfloor$ is prime

5. Positive integers 𝑛 with distance 𝟣 to a perfect square.

6. Positive integers 𝑛 where the number of perfect squares including 𝟢 less than 𝑛 is prime.

7. Prime numbers 𝑛 where either 𝑛 - 𝟤 or 𝑛 + 𝟤 (exclusive) are prime.

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.

9. Positive integers 𝑛 who are the sum of a perfect square and a perfect cube (excluding 𝟢).

10. Positive integers 𝑛 whose decimal digit sum is the cube of a prime.

11. Positive integers 𝑛 for which $\text{decimal\_digitsum}(n)+n$ is a perfect square.

12. Prime numbers 𝑛 for which $\text{decimal\_digitsum}(n^4)$ is prime.

13. Positive integers 𝑛 where $\text{decimal\_digitsum}(2 \cdot n)$ is a substring of 𝑛.

14. Positive integers 𝑛 whose decimal reverse is prime.

15. Positive integers 𝑛 who are a decimal substring of $n^n$.

16. Positive integers 𝑛 whose binary expansion has a prime number of 𝟣’s.

17. Positive integers 𝑛 whose 7-segment representation uses a prime number of segments.