sidef-scripts/Math/sum_of_k-powerful_numbers_in_range.sf

#!/usr/bin/ruby

# Fast recursive algorithm for summing the k-powerful numbers in a given range [A,B].
# A positive integer n is considered k-powerful, if for every prime p that divides n, so does p^k.

# Example:
#   2-powerful = a^2 * b^3,             for a,b >= 1
#   3-powerful = a^3 * b^4 * c^5,       for a,b,c >= 1
#   4-powerful = a^4 * b^5 * c^6 * d^7, for a,b,c,d >= 1

# OEIS:
#   https://oeis.org/A001694 -- 2-powerful numbers
#   https://oeis.org/A036966 -- 3-powerful numbers
#   https://oeis.org/A036967 -- 4-powerful numbers
#   https://oeis.org/A069492 -- 5-powerful numbers
#   https://oeis.org/A069493 -- 6-powerful numbers

# See also:
#   https://oeis.org/A118896 -- Number of powerful numbers <= 10^n.

func powerful_sum_in_range(A, B, k=2) {

    return 0 if (A > B)

    var total = 0

    func (m,r) {

        var from = 1
        var upto = iroot(idiv(B,m), r)

        if (r <= k) {

            if (A > m) {
                # Optimization by Dana Jacobsen (from Math::Prime::Util::PP)
                with (idiv_ceil(A,m)) {|l|
                    if ((l >> r) == 0) {
                        from = 2
                    }
                    else {
                        from = l.iroot(r)
                        from++ if (ipow(from, r) != l)
                    }
                }
            }

            return nil if (from > upto)
            total += m*faulhaber_range(from, upto, r)
            return nil
        }

        each_squarefree(from, upto, {|j|
            j.is_coprime(m) || next
            __FUNC__(m * j**r, r-1)
        })
    }(1, 2*k - 1)

    return total
}

for k in (2..10) {
    var lo = irand(10**(k-1))
    var hi = irand(10**k)
    assert_eq(powerful_sum_in_range(lo, hi, k), k.powerful_sum(lo, hi))
    printf("Sum of %2d-powerful in range 10^j .. 10^(j+1): {%s}\n", k, (7+k).of {|j| powerful_sum_in_range(10**j, 10**(j+1), k) }.join(', '))
}

__END__
Sum of  2-powerful in range 10^j .. 10^(j+1): {22, 502, 19545, 628164, 20656197, 668961441, 21437300251, 685328369991, 21824118507902}
Sum of  3-powerful in range 10^j .. 10^(j+1): {9, 220, 6121, 136410, 3529846, 80934268, 1811337810, 41811161255, 929876351992, 20679545550210}
Sum of  4-powerful in range 10^j .. 10^(j+1): {1, 193, 2493, 60370, 1440893, 26780053, 516891583, 9990376094, 193432085418, 3626702483663, 68456092587576}
Sum of  5-powerful in range 10^j .. 10^(j+1): {1, 96, 1868, 35009, 746121, 14039356, 230448956, 4041417437, 70765409052, 1214243920880, 21187881376824, 365947199216587}
Sum of  6-powerful in range 10^j .. 10^(j+1): {1, 64, 1625, 24108, 427138, 7503765, 142877197, 2128546916, 37085174023, 547117264876, 9207435088386, 149796088225544, 2342746880282546}
Sum of  7-powerful in range 10^j .. 10^(j+1): {1, 0, 896, 24108, 271545, 4519876, 93259499, 1349452792, 22365106723, 310086289407, 4736025082478, 73612282993023, 1102078225069540, 16970183647609915}
Sum of  8-powerful in range 10^j .. 10^(j+1): {1, 0, 768, 21921, 193420, 2016717, 56385643, 851106512, 14014480848, 205584890161, 3186168004038, 43689401756765, 641512327279056, 9291932808199869, 136568208040185109}
Sum of  9-powerful in range 10^j .. 10^(j+1): {1, 0, 512, 15360, 193420, 1626092, 33824682, 596581840, 8827764302, 147389799084, 2165109680321, 29580803725639, 409447338905006, 5697214477371426, 78740331560394730, 1144313243099576141}
Sum of 10-powerful in range 10^j .. 10^(j+1): {1, 0, 0, 15360, 173737, 1626092, 31871557, 284130441, 5610671182, 106206715265, 1591481398917, 21833753103320, 298489744207556, 3892787043427942, 50393901956156445, 725082729912431153, 9766175708618550818}