prog.sf

#!/usr/bin/ruby

# Partial sums of A182139.

# a(n) = Sum_{k=1..n} A069205(floor(n/k)).
# a(n) = Sum_{k=1..n} Sum_{d|k} 2^bigomega(d).
# a(n) = Sum_{k=1..n} 2^bigomega(k) * floor(n/k).

# a(n) = {0, 1, 4, 7, 14, 17, 26, 29, 44, 51, 60, 63, 84, 87, 96, 105, 136, 139, 160, 163, ...}

# Cf. A061503.

# Formulas for: a(n) = Sum_{k=1..n} 2^bigomega(k):
#   a(n) = Sum_{k=1..n} A335073(floor(n/k)).
#   a(n) = Sum_{k=1..n} 2^(bigomega(k) - omega(k)) * floor(n/k).

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

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

say 20.of(f)
say 20.of(a)

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

say 20.of(g)

say 20.of {|n|
    n.divisors.sum{|d|
        2**bigomega(d)
    }
}

func foo(n) {
    sum(1..n, {|k|
        2**bigomega(k)
    })
}

func bar(n) {
    sum(1..n, {|k|
        foo(floor(n/k))
    })
}

func baz(n) {
    sum(1..n, {|k|
        2**bigomega(k) * floor(n/k)
    })
}

say ''

say 20.of(foo)
say 20.of(bar)
say 20.of(baz)