experimental-projects/oeis-research/Daniel Suteu/Liouville sum/new.sf

#!/usr/bin/ruby





func f(n) {
    (1i)**(n**2)
}

func g(n) {
    1i**bigomega(n)
}

func F(n, m) {
    sum(1..n, {|k|
        f(k) * faulhaber(floor(n/k), m)
    })
}

func G(n, m) {
    sum(1..n, {|k|
        g(k) * faulhaber(floor(n/k), m)
    })
}

say 20.of { F(_, 0) }.map{.re}
say 20.of { F(_, 0) }.map{.im}
#say 20.of { G(_, 0) }.map{.im}

__END__
func PF(m) {        # https://oeis.org/A065469
    1e3.primes.prod {|p|
        (1 - 1/(p**m - 1))
    }
}

func PG(m) {            # https://oeis.org/A182448  Pi^2/15 = zeta(2*2) / zeta(2)
    #~ 1e3.primes.prod {|p|
        #~ #(1 - 1/(p**m - 1))
        #~ p**m / (p**m + 1)
    #~ }

    zeta(2*m) / zeta(m)
}

for k in (1..10) {
    #say (G(1e5, k) / faulhaber(1e5, k))
    #say PG(k+1)

    say F(1e4, k)
    say (faulhaber(1e4, k) * PF(k+1))
    say ''
}

#
#say PG(4)


__END__
say 26.of {|n|
    n.divisors.sum{|d|
        f(d)
    } != 0 ? 1 : 0
}.accumulate

say 26.of {|n|
    n.divisors.sum{|d|
        g(d)
    }
}.accumulate



__END__
for k in (1..100) {
    #k.is_powerful && say [k.factor, k.divisors.sum{|d| f(d) }]
    if (k.divisors.sum{|d| g(d) }) {
        say [k.sqrt]
    }
}

__END__
say 20.of {|n|
    n.divisors.sum{|d|
        g(d)
    }
}

__END__

func foo(n) {
    sum(1..n, {|k|
        k.divisors.sum{|d|
            f(k/d) * d
        }
    })
}

func bar(n) {
    n.divisors.sum {|d|
        f(n/d) * d
    }
}

func z(n) {
    n.divisors.sum {|d|
        d.divisors.sum {|d|
            (-1)**omega(d)
        } * euler_phi(n/d)
    }
}

func conj(n) {
    n.factor_prod{|p,e|
        p**e - (p**e - 1)/(p - 1)

        #sum(0..e.dec, {|k|
        #    -p**k
        #})
    }
}

func f2(n) {
    sum(1..n, {|k|
        (-1)**omega(k) * (n//k) * (1 + n//k)
    }) / 2
}

#say 20.of(bar)
for k in (1..500) {
    #say [bar(k), euler_phi(k), euler_phi(k) - bar(k)]
    print(euler_phi(k) - bar(k), ", ")
}

__END__
for n in (1..10000) {

    assert_eq(bar(n), conj(n))

    #var t = 1e30.irand
    #say [z(t), conj(t)]

    #say conj(n.pn_primorial ** 2)
}

#say z(23*7)


__END__
say 20.of(bar).accumulate
say 20.of(f2)

for n in (1..10) {
    say (f2(10**n) / faulhaber(10**n, 1))
}

# sqrt(n)
#say 20.of(foo)
#say 20.of(bar).slice(1)
#say 20.of(foo).map_cons(2, {|a,b| b-a })

#say 800.of(bar)
#~ for k in (1..1000) {
    #~ say bar(k)
#~ }