sum_of_prime-power_exponents_of_factorial.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 15 January 2019
# https://github.com/trizen

# Efficient program for computing the sum of exponents in prime-power factorization of n!.

#~ a(10^1)  = 15
#~ a(10^2)  = 239
#~ a(10^3)  = 2877
#~ a(10^4)  = 31985
#~ a(10^5)  = 343614
#~ a(10^6)  = 3626619
#~ a(10^7)  = 37861249
#~ a(10^8)  = 392351272
#~ a(10^9)  = 4044220058
#~ a(10^10) = 41518796555

# See also:
#   https://oeis.org/A022559
#   https://oeis.org/A071811

func sum_of_exponents_of_factorial (n) {

    return 0 if (n <= 1)

    var s = n.isqrt
    var u = floor(n/(s+1))

    var total = 0
    var prev  = prime_power_count(n)

    for k in (1 .. s) {
        var curr = prime_power_count(floor(n/(k+1)))
        total += k*(prev - curr)
        prev = curr
    }

    for k in (1..u) {
        if (k.is_prime_power) {
            total += floor(n/k)
        }
    }

    return total
}

for k in (1..7) {
    say ("a(10^#{k}) = ", sum_of_exponents_of_factorial(10**k))
}