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A278351 Least number that is the start of a prime-semiprime gap of size n. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

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)

EXAMPLE

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.

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]

PROG

(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

CROSSREFS

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

Sequence in context: A129273 A055988 A275013 * A001075 A293210 A113436

Adjacent sequences:  A278348 A278349 A278350 * A278352 A278353 A278354

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified February 18 05:36 EST 2019. Contains 320245 sequences. (Running on oeis4.)