

A278351


Least number that is the start of a primesemiprime 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
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OFFSET

1,1


COMMENTS

A primesemiprime 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 primesemiprime 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 endofgap 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 primesemiprime 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



