sum_of_remainders.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 10 March 2021
# https://github.com/trizen

# Let's consider the following function:
#   a(n,v) = Sum_{k=1..n} (v mod k)

# The goal is to compute a(n,v) in sublinear time with respect to v.

# Formula:
#   a(n,v) = n*v - A024916(v) + Sum_{k=n+1..v} k*floor(v/k).

# Formula derived from:
#   a(n,v) = Sum_{k=1..n} (v - k*floor(v/k))
#          = n*v - Sum_{k=1..n} k*floor(v/k)
#          = n*v - Sum_{k=1..v} k*floor(v/k) + Sum_{k=n+1..v} k*floor(v/k)

# Related problem:
#   Is there a sublinear formula for computing: Sum_{1<=k<=n, gcd(k,n)=1} k*floor(n/k) ?

# See also:
#   https://oeis.org/A099726 -- Sum of remainders of the n-th prime mod k, for k = 1,2,3,...,n.
#   https://oeis.org/A340976 -- Sum_{1 < k < n} sigma(n) mod k, where sigma = A000203.
#   https://oeis.org/A340180 -- a(n) = Sum_{x in C(n)} (sigma(n) mod x), where C(n) is the set of numbers < n coprime to n, and sigma = A000203.

func T(n) {     # Sum_{k=1..n} k = n-th triangular number
    n.faulhaber(1)
}

func S(n) {     # A024916(n) = Sum_{k=1..n} sigma(k) = Sum_{k=1..n} k*floor(n/k)
    #with (n.isqrt) { |s| sum(1..s, {|k| T(idiv(n,k)) + k*idiv(n,k) }) - T(s)*s }

    if (n < 0) {
        return (T(n.abs) + __FUNC__(n.abs-1))
    }

    n.dirichlet_sum({1}, {_}, {_}, {.faulhaber(1)})
}

func g(a,b) {   # g(a,b) = Sum_{k=a..b} k*floor(b/k)

    if (b < 0) {
        return (T(b.abs) - T(a.abs-1)  + __FUNC__(a, b.abs-1))
    }

    var total = 0
    while (a <= b) {
        var t = idiv(b, a)
        var u = idiv(b, t)
        total += t*(T(u) - T(a-1))
        a = u+1
    }
    return total
}

func sum_remainders(n, v) {     # sub-linear formula
    sgn(v) * (n*v.abs - S(v) + g(n+1, v))
}

say {|n| sum_remainders(n,   n.prime) }.map(1..20)        #=> A099726
say {|n| sum_remainders(n-1, n.sigma) }.map(1..20)        #=> A340976

for k in (1..8) {
    say ("A099726(10^#{k}) = ", sum_remainders(10**k, prime(10**k)))
}

# Positive v
assert_eq(
    20.of {|n| 20.of {|v| sum(1..n, {|k| v % k }) } },
    20.of {|n| 20.of {|v| sum_remainders(n,v) } }
)

# Negative v
assert_eq(
    20.of {|n| 20.of {|v| sum(1..n, {|k| -v % k }) } },
    20.of {|n| 20.of {|v| sum_remainders(n,-v) } }
)

__END__
A099726(10^1) = 30
A099726(10^2) = 2443
A099726(10^3) = 248372
A099726(10^4) = 25372801
A099726(10^5) = 2437160078
A099726(10^6) = 252670261459
A099726(10^7) = 24690625139657
A099726(10^8) = 2516604108737704