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- #!/usr/bin/perl
- # Author: Daniel "Trizen" Șuteu
- # Date: 10 November 2018
- # https://github.com/trizen
- # A new generalized algorithm with O(sqrt(n)) complexity for computing the partial-sums of `k * sigma_j(k)`, for `1 <= k <= n`:
- #
- # Sum_{k=1..n} k * sigma_j(k)
- #
- # for any integer j >= 0.
- # Example: `a(n) = Sum_{k=1..n} k * sigma(k)`
- # a(10^1) = 622
- # a(10^2) = 558275
- # a(10^3) = 549175530
- # a(10^4) = 548429473046
- # a(10^5) = 548320905633448
- # a(10^6) = 548312690631798482
- # a(10^7) = 548311465139943768941
- # a(10^8) = 548311366911386862908968
- # a(10^9) = 548311356554322895313137239
- # a(10^10) = 548311355740964925044531454428
- # For m>=0 and j>=1, we have the following asymptotic formula:
- # Sum_{k=1..n} k^m * sigma_j(k) ~ zeta(j+1)/(j+m+1) * n^(j+m+1)
- # See also:
- # https://en.wikipedia.org/wiki/Divisor_function
- # https://en.wikipedia.org/wiki/Faulhaber%27s_formula
- # https://en.wikipedia.org/wiki/Bernoulli_polynomials
- # https://trizenx.blogspot.com/2018/11/partial-sums-of-arithmetical-functions.html
- use 5.020;
- use strict;
- use warnings;
- use ntheory qw(divisors);
- use experimental qw(signatures);
- use Math::AnyNum qw(faulhaber_sum sum isqrt ipow);
- sub sigma_partial_sum($n, $m) { # O(sqrt(n)) complexity
- my $total = 0;
- my $s = isqrt($n);
- my $u = int($n / ($s + 1));
- for my $k (1 .. $s) {
- $total += $k*($k+1) * (faulhaber_sum(int($n/$k), $m+1) - faulhaber_sum(int($n/($k+1)), $m+1));
- }
- for my $k (1 .. $u) {
- $total += ipow($k, $m+1) * int($n/$k) * (1 + int($n/$k));
- }
- return $total/2;
- }
- sub sigma_partial_sum_test($n, $m) { # just for testing
- sum(map { $_ * sum(map { ipow($_, $m) } divisors($_)) } 1..$n);
- }
- for my $m (0..10) {
- my $n = int(rand(1000));
- my $t1 = sigma_partial_sum($n, $m);
- my $t2 = sigma_partial_sum_test($n, $m);
- die "error: $t1 != $t2" if ($t1 != $t2);
- say "Sum_{k=1..$n} k * σ_$m(k) = $t2"
- }
- __END__
- Sum_{k=1..649} k * σ_0(k) = 1505437
- Sum_{k=1..184} k * σ_1(k) = 3442689
- Sum_{k=1..156} k * σ_2(k) = 180861250
- Sum_{k=1..781} k * σ_3(k) = 63090289257686
- Sum_{k=1..822} k * σ_4(k) = 53514505511600484
- Sum_{k=1..982} k * σ_5(k) = 128445772086331164364
- Sum_{k=1..742} k * σ_6(k) = 11644176895188820029668
- Sum_{k=1..837} k * σ_7(k) = 22614022054863154308526282
- Sum_{k=1..355} k * σ_8(k) = 3230297764819153302018985
- Sum_{k=1..837} k * σ_9(k) = 12937980446016909148074821860258
- Sum_{k=1..699} k * σ_10(k) = 1144140317656849776081892799180303
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