A174641 Smallest prime that begins a run of n consecutive primes that are not Ramanujan primes.

3, 3, 3, 73, 191, 191, 509, 2539, 2539, 5279, 9901, 9901, 9901, 11593, 11593, 55343, 55343, 55343, 55343, 55343, 174929, 174929, 174929, 225977, 225977, 225977, 225977, 225977, 534889, 534889, 534889, 534889, 534889, 534889, 534889, 534889, 2492317, 2492317

Offset

1,1

Comments

The run of 10 consecutive non-Ramanujan primes was mentioned by Sondow.

Links

Dana Jacobsen, Table of n, a(n) for n = 1..107 (first 67 terms from T. D. Noe)

J. Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-2010.

J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 (2009) 630-635.

J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011.

J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2.

Mathematica

nn=10000; t=Table[0, {nn}]; len=Prime[3*nn]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s<nn, t[[s+1]]=k], {k, len}]; t=t+1; t=Complement[Prime[Range[PrimePi[t[[-1]]]]], t]; ind=PrimePi[t]; d=Differences[ind]; cnt=0; n=1; Join[{2}, Reap[Do[If[d[[i]]==1, cnt++; If[cnt==n, Sow[t[[i-n+1]]]; n++], cnt=0], {i, Length[d]}]][[2, 1]]]

Prog

(Perl) use ntheory ":all";
my($k, $max, $start, $end, $inc, $p, $q, $r, $pi)
   = (0, 0, 0, 10, 1e9, 0, 2, [], prime_iterator(3));
while (1) {
  if (!@$r) {
    ($start, $end) = ($end+1, $end+$inc);
    $r = ramanujan_primes($start, $end);
  }
  ($p, $q, $k) = ($q, shift(@$r), 0);
  # $k = prime_count($p+1, $q-1);
  $k++ while $pi->() < $q;
  say ++$max, " ", next_prime($p)   while $k > $max;
}
# Dana Jacobsen, Jul 14 2016

Crossrefs

Cf. A104272 (Ramanujan primes), A174635 (non-Ramanujan primes).

Cf. A174602 (runs of Ramanujan primes).

Cf. A202187, A202188.

Sequence in context: A174538 A340821 A091323 * A217671 A118539 A015665

Adjacent sequences: A174638 A174639 A174640 * A174642 A174643 A174644

Keyword

nonn

Author

T. D. Noe, Nov 29 2010

Status

approved