

A174602


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


6



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
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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], 20092010.
J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 (2009) 630635.
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[[in+1]]]; n++], cnt=0], {i, Length[d]}]][[2, 1]]]


PROG

(Perl) use ntheory ":all"; my $r=ramanujan_primes(1e8); my $max = 0; for (0..$#$r2) { 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 nonRamanujan primes).
Sequence in context: A275114 A217599 A107214 * A154880 A160958 A046848
Adjacent sequences: A174599 A174600 A174601 * A174603 A174604 A174605


KEYWORD

nonn


AUTHOR

T. D. Noe, Nov 29 2010


STATUS

approved



