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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, 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 \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\}
  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\}
Source code: seventeen.py
Jonathan Frech's blog; built 2024/03/18 18:45:40 CET