A174602 Smallest prime that begins a run of n Ramanujan primes that are consecutive primes.

2, 67, 227, 227, 227, 2657, 2657, 2657, 2657, 2657, 2657, 2657, 2657, 562871, 793487, 809707, 809707, 984241, 984241, 984241, 6234619, 11652013, 41662651, 41662651, 41662651, 94653397, 383825567, 869730887, 953913871, 953913871, 953913871

Offset

1,1

Comments

The first run of 13 consecutive Ramanujan primes was mentioned by Sondow.

Starting at index m = A191228(a(n)) in A190874(m), the first instance of a count of n - 1 consecutive 1's is seen. - John W. Nicholson, Dec 15 2011

Links

Dana Jacobsen, Table of n, a(n) for n = 1..42

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.

Example

67 and 71 are the first two Ramanujan primes that are consecutive primes, so a(2) = 67.

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; 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 $r=ramanujan_primes(1e8); my $max = 0; for (0..$#$r-2) { my $k=0; $k++ while next_prime($r->[$_+$k]) == $r->[$_+$k+1]; say ++$max, " ", $r->[$_] while $k >= $max; } # Dana Jacobsen, Jul 14 2016

Crossrefs

Cf. A104272 (Ramanujan primes), A174641 (runs of non-Ramanujan primes).

Sequence in context: A217599 A107214 A371509 * A154880 A160958 A046848

Adjacent sequences: A174599 A174600 A174601 * A174603 A174604 A174605

Keyword

nonn

Author

T. D. Noe, Nov 29 2010

Status

approved