A181678 - OEIS

%I #23 Sep 06 2015 07:40:59

%S 0,0,10,73,508,3468,25629,194614,1537504,12447679,102834428

%N Number of twin Ramanujan prime pairs less than 10^n.

%H J. Sondow, <a href="http://arxiv.org/abs/0907.5232"> Ramanujan primes and Bertrand's postulate</a>, Amer. Math. Monthly 116 (2009) 630-635.

%H J. Sondow, J. W. Nicholson, and T. D. Noe, <a href="http://arxiv.org/abs/1105.2249"> Ramanujan Primes: Bounds, Runs, Twins, and Gaps</a>, J. Integer Seq. 14 (2011) Article 11.6.2

%t nn=50000; t=Table[0,{nn}]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s<nn, t[[s+1]]=k], {k, Prime[3*nn]}]; t=t+1; cnt=0; i=1; Table[While[t[[i]]<10^n, If[t[[i]]+2 == t[[i+1]], cnt++]; i++]; cnt, {n,Floor[Log[10,t[[-1]]]]}]

%o (Perl) use ntheory ":all"; for my $n (0..9) { my $r = ramanujan_primes(10**$n); my $t=0; for (0..$#$r-1) { $t++ if $r->[$_+1] - $r->[$_] == 2; } say "$n $t"; } # _Dana Jacobsen_, Sep 06 2015

%Y Cf. A178127 (lesser of twin Ramanujan primes), A007508 (number of twin primes pairs < 10^n), A190654 (twin Ramanujan primes).

%K nonn,more

%O 1,3

%A _T. D. Noe_, Nov 18 2010

%E a(10)-a(11) from _Dana Jacobsen_, Apr 29 2015