A278349 a(n) is the smallest semiprime followed by gap (to the next semiprime) equal to n-th semiprime.

10, 15, 146, 237, 597, 1603, 14545, 13863, 16053, 55805, 393565, 219509, 153263, 35981, 1789339, 4686907, 11047438, 7487726, 1589662, 39291566, 12581421, 6974465, 93519854, 140140538, 5835191, 756869018, 251306317, 285074689, 1102062347, 327023206, 4282236806

Offset

1,1

Comments

It appears that a(n) exists for any n.

a(41) = 42264532659, a(42) = 34279038379, a(45) = 58898255693, a(47) = 70265467489, and a(50) = 49167925231. - Charles R Greathouse IV, Dec 07 2016

Links

Dana Jacobsen, Table of n, a(n) for n = 1..55 (first 39 terms from Charles R Greathouse IV)

Example

a(1)=10 because p=10 and q=14 are two consecutive semiprimes and q-p=4 (first semiprime A001358(1)),
a(2)=15 because p=15 and q=21 are two consecutive semiprimes and q-p=6 (2nd semiprime A001358(2)),
a(3)=146 because p=146 and q=155 are two consecutive semiprimes and q-p=6=A001358(3)).

Prog

(PARI) issemi(n)=bigomega(n)==2
nextsemi(n)=while(!issemi(n), n++); n
listsemi(a, b)=my(v=List()); if(a<4, a=4); forprime(p=sqrtint(a-1)+1, b\2, forprime(q=(a-1)\p+1, min(b\p, p), listput(v, p*q))); Set(v)
list(lim)=
{
  lim=nextsemi(ceil(lim));
  my(foundsemi=[0], small=listsemi(4, max(sqrtint(lim), 4)), v=small, lastsmall=small[#small], needed=4);
  while(v[1]<lim,
    for(i=2, #v,
      my(g=v[i]-v[i-1], idx);
      if(g<needed, next);
      if(g>lastsmall,
        small=listsemi(4, max(2*lastsmall, g));
        lastsmall=small[#small]
      );
      idx=setsearch(small, g);
      if(!idx, next);
      if(idx>#foundsemi,
        foundsemi=concat(foundsemi, vector(idx-#foundsemi));
      );
      if(foundsemi[idx]==0,
        foundsemi[idx]=v[i-1];
        if(needed==g,
          for(j=idx, #foundsemi,
            if(foundsemi[j]==0, break);
            needed=small[j+1];
          )
        );
      )
    );
    v=listsemi(v[#v], min(nextsemi(v[#v]+10^7), lim));
  );
  for(i=1, #foundsemi,
    if(foundsemi[i]==0, return(foundsemi[1..i-1]))
  );
  foundsemi;
} \\ Charles R Greathouse IV, Nov 19 2016
(Perl) use ntheory ":all";
  my($l, $nxti, $nxt, @G)=(4, 1, 4);
  forsemiprimes {
    if (!exists $G[$_-$l]) {
      $G[$_-$l]=$l;
      while (exists $G[$nxt]) {
        print "$nxti $G[$nxt]\n";
        $nxt=nth_semiprime(++$nxti);
      }
    }
    $l=$_;
} 5, 1e8; # Dana Jacobsen, Oct 16 2018

Crossrefs

Cf. A001358.

Sequence in context: A339314 A166626 A238759 * A114703 A134515 A232608

Adjacent sequences: A278346 A278347 A278348 * A278350 A278351 A278352

Keyword

nonn

Author

Zak Seidov, Nov 19 2016

Extensions

a(14) and a(19) corrected by Charles R Greathouse IV, Nov 19 2016

a(27)-a(35) from Charles R Greathouse IV, Nov 19 2016

a(36)-a(38) from Charles R Greathouse IV, Nov 22 2016

Status

approved