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- #!/usr/bin/perl
- # a(n) is the least k such that k^2 + 1 is a semiprime p*q, p < q, and (q - p)/2^n is prime.
- # https://oeis.org/A258780
- # New terms found:
- # a(20) = 126072244
- # a(21) = 9586736
- # a(22) = 4156840
- # a(23) = 542759984 # to be added
- # a(24) = 1017981724 # to be added
- # a(25) = 2744780140 # to be added
- # a(26) = 405793096 # to be added
- # a(27) = 148647496 # to be added
- # a(28) = 1671024916 # to be added
- # ....
- # a(30) = 1515257276 # to be added
- # Another example for a(27) is 2808502256.
- # It seems that all terms are multiples of 4.
- use 5.014;
- use ntheory qw(:all);
- sub f {
- my %seen;
- foreach my $n (1 .. 1e9) {
- my $k = $n << 1; # all terms are even
- my $t = $k * $k + 1;
- if (is_semiprime($t)) {
- my ($p, $q) = factor($t);
- my $v = valuation($q - $p, 2);
- #$v > 22 or next;
- if (not exists $seen{$v - 1} and ((1<<$v) == $q - $p)) {
- $seen{$v - 1} = 1;
- say $v-1, " => $k,";
- }
- elsif (not exists $seen{$v} and is_prime(($q - $p) >> $v)) {
- $seen{$v} = 1;
- say "$v => $k,";
- }
- }
- }
- }
- f();
- __END__
- 2 => 8,
- 3 => 12,
- 5 => 64,
- 4 => 140,
- 7 => 196,
- 8 => 1300,
- 9 => 1600,
- 6 => 2236,
- 11 => 5084,
- 10 => 6256,
- 15 => 36680,
- 19 => 104120,
- 13 => 246196,
- 12 => 248756,
- 14 => 484400,
- 17 => 821836,
- 16 => 887884,
- 18 => 1559116,
- 22 => 4156840,
- 21 => 9586736,
- 20 => 126072244,
- 27 => 148647496,
- 26 => 405793096,
- 23 => 542759984,
- 24 => 1017981724,
- 30 => 1515257276,
- 28 => 1671024916,
- 25 => 2744780140,
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