prime_omega_function_generalized.sf

#!/usr/bin/ruby

# Author: Daniel "Trizen" Șuteu
# Date: 25 November 2018
# https://github.com/trizen

# Generalization of the prime omega function `ω_m(n)`, for `m>=0`:
#   ω_m(n) = n^m * Sum_{p|n} 1/p^m

# Where the standard prime omega function `ω(n)` is equivalent with `ω_0(n)`.

# See also:
#   https://oeis.org/A069359
#   https://oeis.org/A322068
#   https://en.wikipedia.org/wiki/Prime_omega_function
#   https://trizenx.blogspot.com/2018/08/interesting-formulas-and-exercises-in.html

func omega(n, m) {
    n**m * n.factor_exp.sum {|p| 1 / p[0]**m }
}

say 25.of { omega(_, 0) }
say 25.of { omega(_, 1) }
say 25.of { omega(_, 2) }
say 25.of { omega(_, 3) }
say 25.of { omega(_, 4) }
say 25.of { omega(_, 5) }

__END__
[0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2]
[0, 0, 1, 1, 2, 1, 5, 1, 4, 3, 7, 1, 10, 1, 9, 8, 8, 1, 15, 1, 14, 10, 13, 1, 20]
[0, 0, 1, 1, 4, 1, 13, 1, 16, 9, 29, 1, 52, 1, 53, 34, 64, 1, 117, 1, 116, 58, 125, 1, 208]
[0, 0, 1, 1, 8, 1, 35, 1, 64, 27, 133, 1, 280, 1, 351, 152, 512, 1, 945, 1, 1064, 370, 1339, 1, 2240]
[0, 0, 1, 1, 16, 1, 97, 1, 256, 81, 641, 1, 1552, 1, 2417, 706, 4096, 1, 7857, 1, 10256, 2482, 14657, 1, 24832]
[0, 0, 1, 1, 32, 1, 275, 1, 1024, 243, 3157, 1, 8800, 1, 16839, 3368, 32768, 1, 66825, 1, 101024, 17050, 161083, 1, 281600]