| 1234567891011121314151617181920212223242526272829303132333435363738 |
- #!/usr/bin/ruby
- # Möbius inversion formula.
- # See also:
- # https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula
- func moebius_transform(n, f={.sigma}) {
- n.divisor_sum {|d|
- moebius(d) * f(n/d)
- }
- }
- func inverse_moebius_transform(n, f={.sigma}) {
- n.divisor_sum {|d|
- f(d)
- }
- }
- say "=> Möbius transform of Jordan's J_2(k) totient function:"
- say 20.of { moebius_transform(_, { .jordan_totient(2) }) } # Möbius transform applied once to the Jordan J_2(k) function.
- say 20.of { moebius_transform(_, { moebius_transform(_, { _**2 }) }) } # Möbius transform applied twice to squares.
- say "\n=> Inverse Möbius transform of sigma_2(k) function:"
- say 20.of { inverse_moebius_transform(_, { .sigma(2) }) } # Inverse Möbius transform applied once to the sigma_2(k) function.
- say 20.of { inverse_moebius_transform(_, { inverse_moebius_transform(_, { _**2 }) }) } # Inverse Möbius transform applied twice to squares.
- __END__
- => Möbius transform of Jordan's J_2(k) totient function:
- [0, 1, 2, 7, 9, 23, 14, 47, 36, 64, 46, 119, 63, 167, 94, 161, 144, 287, 128, 359]
- [0, 1, 2, 7, 9, 23, 14, 47, 36, 64, 46, 119, 63, 167, 94, 161, 144, 287, 128, 359]
- => Inverse Möbius transform of sigma_2(k) function:
- [0, 1, 6, 11, 27, 27, 66, 51, 112, 102, 162, 123, 297, 171, 306, 297, 453, 291, 612, 363]
- [0, 1, 6, 11, 27, 27, 66, 51, 112, 102, 162, 123, 297, 171, 306, 297, 453, 291, 612, 363]
|
