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prime_big_omega_function_generalized_old.sf

prime_big_omega_function_generalized_old.sf 1.4 KB
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  1. #!/usr/bin/ruby
    2. 

  2. 

  3. # Author: Daniel "Trizen" Șuteu
    4. 

  4. # Date: 25 November 2018
    5. 

  5. # https://github.com/trizen
    6. 

  6. 

  7. # Generalization of the prime big omega function `Ω_m(n)`, for `m>=0`:
    8. 

  8. #   Ω_m(n) = n^m * Sum_{p^k|n} k/p^m
    9. 

  9. 

  10. # Where the standard prime big omega function `Ω(n)` is equivalent with `Ω_0(n)`.
    11. 

  11. 

  12. # See also:
    13. 

  13. #   https://oeis.org/A001222
    14. 

  14. #   https://oeis.org/A022559
    15. 

  15. #   https://oeis.org/A003415
    16. 

  16. #   https://oeis.org/A190121
    17. 

  17. #   https://en.wikipedia.org/wiki/Prime_omega_function
    18. 

  18. #   https://trizenx.blogspot.com/2018/08/interesting-formulas-and-exercises-in.html
    19. 

  19. 

  20. func bigomega(n, m) {
    21. 

  21.     n**m * n.factor_exp.sum {|p| p[1] / p[0]**m }
    22. 

  22. }
    23. 

  23. 

  24. say 25.of { bigomega(_, 0) }
    25. 

  25. say 25.of { bigomega(_, 1) }
    26. 

  26. say 25.of { bigomega(_, 2) }
    27. 

  27. say 25.of { bigomega(_, 3) }
    28. 

  28. say 25.of { bigomega(_, 4) }
    29. 

  29. say 25.of { bigomega(_, 5) }
    30. 

  30. 

  31. __END__
    32. 

  32. [0, 0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 4]
    33. 

  33. [0, 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, 8, 32, 1, 21, 1, 24, 10, 13, 1, 44]
    34. 

  34. [0, 0, 1, 1, 8, 1, 13, 1, 48, 18, 29, 1, 88, 1, 53, 34, 256, 1, 153, 1, 216, 58, 125, 1, 496]
    35. 

  35. [0, 0, 1, 1, 16, 1, 35, 1, 192, 54, 133, 1, 496, 1, 351, 152, 2048, 1, 1161, 1, 2064, 370, 1339, 1, 5696]
    36. 

  36. [0, 0, 1, 1, 32, 1, 97, 1, 768, 162, 641, 1, 2848, 1, 2417, 706, 16384, 1, 9153, 1, 20256, 2482, 14657, 1, 66304]
    37. 

  37. [0, 0, 1, 1, 64, 1, 275, 1, 3072, 486, 3157, 1, 16576, 1, 16839, 3368, 131072, 1, 74601, 1, 201024, 17050, 161083, 1, 779264]
    38. 

  38.