A278351 Least number that is the start of a prime-semiprime gap of size n.

2, 7, 26, 97, 341, 241, 6091, 3173, 2869, 2521, 16022, 26603, 114358, 41779, 74491, 39343, 463161, 104659, 248407, 517421, 923722, 506509, 1930823, 584213, 2560177, 4036967, 4570411, 4552363, 7879253, 4417813, 27841051, 5167587, 13683034, 9725107, 47735342, 25045771, 63305661

Offset: 1

Author

Bobby Jacobs, Charles R Greathouse IV, Jonathan Vos Post, and Robert G. Wilson v, Nov 23 2016

Keywords

• nonn

Comments

A prime-semiprime gap of n is defined as the difference between p & q, p being either a prime, A000040, or a semiprime, A001358, and q being the next greater prime or semiprime, see examples.

The corresponding numbers at the end of the prime-semiprime gaps, i.e., a(n)+n, are in A278404.

In the first 52 terms, 19 are primes and the remaining 33 are semiprime. Of the end-of-gap terms a(n)+n, 20 are primes and 32 are not. There are only 6 pairs of p and q that are both primes, and 19 pairs that are both semiprime.

Examples

			a(1) = 2 since there is a gap of 1 between 2 and 3, both of which are primes.
a(2) = 7 since there is a gap of 2 between 7 and 9, the first is a prime and the second is a semiprime.
a(3) = 26 since there is a gap of 3 between 26, a semiprime, and 29, a prime.
a(6) = 241 because the first prime-semiprime gap of size 6 is between 241 and 247.
		

Links

• Dana Jacobsen, Table of n, a(n) for n = 1..106 (first 52 terms from Bobby Jacobs, Charles R Greathouse IV, Jonathan Vos Post and Robert G. Wilson v)

Crossrefs

Cf. A000230, A037143, A131109, A275108, A275013, A275014, A278404.

Programs

Mathematica

nxtp[n_] := Block[{m = n + 1}, While[ PrimeOmega[m] > 2, m++]; m]; gp[_] = 0; p = 2; While[p < 1000000000, q = nxtp[p]; If[ gp[q - p] == 0, gp[q -p] = p; Print[{q -p, p}]]; p = q]; Array[gp, 40]

Perl

use ntheory ":all";
my($final,$p,$nextn,@gp) = (40,2,1);  # first 40 values in order
forfactored {
  if (scalar(@) <= 2) { my $q = $;
    if (!defined $gp[$q-$p]) {
      $gp[$q-$p] = $p;
      while ($nextn <= $final && defined $gp[$nextn]) {
        print "$nextn $gp[$nextn]\n";
        $nextn++;
      }
      lastfor if $nextn > $final;
    }
    $p = $q;
  }
} 3,10**14; # Dana Jacobsen, Sep 10 2018