moebius_transform.sf

#!/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]