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- #!/usr/bin/perl
- # Numbers that are not powers of primes (A024619) whose harmonic mean of their proper unitary divisors is an integer.
- # https://oeis.org/A335270
- # Known terms:
- # 228, 1645, 7725, 88473, 20295895122, 22550994580
- # Conjecture: all terms have the form n*(usigma(n)-1) where usigma(n)-1 is prime.
- # The conjecture was inspired by the similar conjecture of Chai Wah Wu from A247077.
- use 5.014;
- use strict;
- #use integer;
- use ntheory qw(:all);
- sub usigma {
- vecprod(map { powint($_->[0], $_->[1]) + 1 } factor_exp($_[0]));
- }
- my $count = 0;
- foreach my $k (2 .. 1e9) {
- my $p = usigma($k) - 1;
- is_prime($p) || next;
- my $m = mulint($k, $p);
- next if ($k == $p);
- my $o = prime_omega($k) + 1;
- if (++$count >= 1e5) {
- say "Testing: $k -> $m";
- $count = 0;
- }
- if (modint(mulint($m, ((1 << $o) - 1)), mulint(usigma($k), $p+1) - 1) == 0) {
- say "\tFound: $k -> $m";
- die "New term: $k -> $m\n" if ($m > 22550994580);
- }
- }
- __END__
- Found: 12 -> 228
- Found: 35 -> 1645
- Found: 75 -> 7725
- Found: 231 -> 88473
- Found: 108558 -> 20295895122
- Found: 120620 -> 22550994580
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