prog.sf

#!/usr/bin/ruby

# Conjecture: partial sums of A298826. (with changed sign for each second term)

# a(n) = Sum_{k=1..n} A174863(floor(n/k)).
# a(n) = Sum_{k=1..n} Sum_{d|k} (-1)^omega(d).
# a(n) = Sum_{k=1..n} (-1)^omega(k) * floor(n/k).

# Conjecture: a(n) = Sum_{k=1..n} (-1)^(k+1) * A298826(k).

# Terms:
#   1, 1, 1, 0, 0, 0, 0, -2, -3, -3, -3, -3, -3, -3, -3, -6, -6, -6, -6, -6, -6, -6, -6, -6, -7, -7, -9, -9, -9, -9, -9, -13, -13, -13, -13, -12, -12, -12, -12, -12, -12, -12, -12, -12, -12, -12, -12, -12, -13, -13, -13, -13, -13, -13, -13, -13, -13, -13, -13

# OEIS:
#   https://oeis.org/A298826  -- a(n) = A298825(n)/n.
#   https://oeis.org/A174863  -- Little omega analog to Liouville's function L(n).
#   https://oeis.org/A306408  -- a(n) = Sum_{d|n} (-1)^omega(n/d) * d.

func f(n) {
    n.divisors.sum{|d|
        (-1)**omega(d)
    }
}

func A174863(n) {
    sum(1..n, {|k|
        (-1)**omega(k)
    })
}

func a(n) {
    sum(1..n, {|k|
        A174863(floor(n/k))
    })
}

func a2(n) {
    sum(1..n, {|k|
        f(k)
    })
}

func a3(n) {
    sum(1..n, {|k|
        (-1)**omega(k) * floor(n/k)
    })
}

say 30.of(a)
say 30.of(a2)
say 30.of(a3)

__END__


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

say 100.of(f2).map_cons(2,{|a,b| b-a })

__END__

#say 20.of(f2).accumulate
500.of(f2).each{|k|
    say k

}

__END__
#say 200.of(f2)

func a(n) {
    #~ n.factor_prod{|p,e|
         #~ p**e - (p**e - 1)/(p-1)
    #~ }
    1..n -> sum{|k|
        #k.moebius
        (-1)**omega(k)
    }
}

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

func c(n) {
    n * n.divisors.sum{|d|
        (-1)**bigomega(d) / d
    }
}

say 20.of(a)
say 20.of(b)
say 20.of(c)

func f(n) {
    sum(1..n, {|k|
        a(floor(n/k))
    })
}

func g(n) {
    sum(1..n, {|k|
        b(floor(n/k))
    })
}

func h(n) {
    sum(1..n, {|k|
        c(floor(n/k))
    })
}

say ''

say 20.of(f)
say 20.of(g)
say 20.of(h)

#say f(1000)
#say f(1000)

#say sum(1..1000, {|d|
#    d.sigma0
#})