Jonathan. Frech’s WebBlog

Seventeen (#174)

Jonathan Frech

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.

    $$\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, \dots\}$$$$\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, \dots\}$$

  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.

    $$\{5, 7, 15, 17, 19, 23, 25, 35, 75, 117, 177, 207, 225, 237, 321, 393, 453, 495, 555, 567, \dots\}$$$$\{5, 7, 15, 17, 19, 23, 25, 35, 75, 117, 177, 207, 225, 237, 321, 393, 453, 495, 555, 567, \dots\}$$

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

    $$\{5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 50, 51, 52, \dots\}$$$$\{5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 50, 51, 52, \dots\}$$

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

    $$\{4, 5, 6, 7, 8, 12, 13, 17, 18, 19, 28, 29, 33, 34, 35, 44, 45, 46, 49, 50, \dots\}$$$$\{4, 5, 6, 7, 8, 12, 13, 17, 18, 19, 28, 29, 33, 34, 35, 44, 45, 46, 49, 50, \dots\}$$

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

    $$\{1, 2, 3, 5, 8, 10, 15, 17, 24, 26, 35, 37, 48, 50, 63, 65, 80, 82, 99, 101, \dots\}$$$$\{1, 2, 3, 5, 8, 10, 15, 17, 24, 26, 35, 37, 48, 50, 63, 65, 80, 82, 99, 101, \dots\}$$

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

    $$\{2, 3, 4, 5, 6, 7, 8, 9, 17, 18, 19, 20, 21, 22, 23, 24, 25, 37, 38, 39, \dots\}$$$$\{2, 3, 4, 5, 6, 7, 8, 9, 17, 18, 19, 20, 21, 22, 23, 24, 25, 37, 38, 39, \dots\}$$

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

    $$\{3, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, 137, 139, \dots\}$$$$\{3, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, 137, 139, \dots\}$$

  8. Positive integers 𝑛 whose three-di­men­sion­al vector’s $(n,n,n)$$(n,n,n)$ floored length is prime, $\lfloor\sqrt{3\cdot n^2}\rfloor$$\lfloor\sqrt{3\cdot n^2}\rfloor$ is prime.

    $$\{2, 3, 8, 10, 11, 17, 18, 24, 25, 31, 39, 41, 46, 48, 60, 62, 63, 76, 91, 100, \dots\}$$$$\{2, 3, 8, 10, 11, 17, 18, 24, 25, 31, 39, 41, 46, 48, 60, 62, 63, 76, 91, 100, \dots\}$$

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

    $$\{2, 5, 9, 10, 12, 17, 24, 26, 28, 31, 33, 36, 37, 43, 44, 50, 52, 57, 63, 65, \dots\}$$$$\{2, 5, 9, 10, 12, 17, 24, 26, 28, 31, 33, 36, 37, 43, 44, 50, 52, 57, 63, 65, \dots\}$$

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

    $$\{8, 17, 26, 35, 44, 53, 62, 71, 80, 107, 116, 125, 134, 143, 152, 161, 170, 206, 215, 224, \dots\}$$$$\{8, 17, 26, 35, 44, 53, 62, 71, 80, 107, 116, 125, 134, 143, 152, 161, 170, 206, 215, 224, \dots\}$$

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

    $$\{2, 8, 17, 27, 38, 72, 86, 135, 161, 179, 216, 245, 275, 315, 347, 432, 467, 521, 558, 614, \dots\}$$$$\{2, 8, 17, 27, 38, 72, 86, 135, 161, 179, 216, 245, 275, 315, 347, 432, 467, 521, 558, 614, \dots\}$$

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

    $$\{2, 5, 7, 17, 23, 41, 47, 53, 67, 73, 97, 103, 113, 151, 157, 163, 173, 179, 197, 199, \dots\}$$$$\{2, 5, 7, 17, 23, 41, 47, 53, 67, 73, 97, 103, 113, 151, 157, 163, 173, 179, 197, 199, \dots\}$$

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

    $$\{9, 17, 25, 52, 58, 66, 71, 85, 90, 104, 107, 115, 118, 123, 137, 142, 151, 156, 165, 170, \dots\}$$$$\{9, 17, 25, 52, 58, 66, 71, 85, 90, 104, 107, 115, 118, 123, 137, 142, 151, 156, 165, 170, \dots\}$$

  14. Positive integers 𝑛 whose decimal reverse is prime.

    $$\{2, 3, 5, 7, 11, 13, 14, 16, 17, 20, 30, 31, 32, 34, 35, 37, 38, 50, 70, 71, \dots\}$$$$\{2, 3, 5, 7, 11, 13, 14, 16, 17, 20, 30, 31, 32, 34, 35, 37, 38, 50, 70, 71, \dots\}$$

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

    $$\{1, 5, 6, 9, 10, 11, 16, 17, 19, 21, 24, 25, 28, 31, 32, 33, 35, 36, 37, 39, \dots\}$$$$\{1, 5, 6, 9, 10, 11, 16, 17, 19, 21, 24, 25, 28, 31, 32, 33, 35, 36, 37, 39, \dots\}$$

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

    $$\{3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, \dots\}$$$$\{3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, \dots\}$$

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

    $$\{1, 2, 3, 5, 7, 8, 12, 13, 15, 17, 20, 21, 26, 29, 30, 31, 36, 39, 47, 48, \dots\}$$$$\{1, 2, 3, 5, 7, 8, 12, 13, 15, 17, 20, 21, 26, 29, 30, 31, 36, 39, 47, 48, \dots\}$$

Source code: seventeen.py