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A273257 Number of twin primes between prime(n) and prime(n)^2.

%I A273257 #34 Jul 04 2019 03:42:50
%S A273257 0,1,3,4,8,9,16,17,21,29,30,41,48,50,61,74,87,91,110,121,123,138,152,
%T A273257 166,187,202,208,218,223,234,276,288,315,320,365,374,394,411,432,455,
%U A273257 480,492,541,547,567,574,626,685,708,716,732,764,772,818,851
%N A273257 Number of twin primes between prime(n) and prime(n)^2.
%C A273257 Both p and p+2 must appear in the indicated range, and a prime can only be used once (so (3, 5) and (5, 7) can't both be used).
%C A273257 It appears that there should be more twin primes between prime(n) and prime(n)^2 as n increases. Specifically this sequence should be strictly increasing.
%C A273257 Indeed even the number of twin primes between prime(n)^2 and prime(n+1)^2 (A057767) seems to have a lower bound of about n/11. - _M. F. Hasler_, Jun 27 2019
%H A273257 Charles R Greathouse IV, <a href="/A273257/b273257.txt">Table of n, a(n) for n = 1..10000</a>
%e A273257 For n=3, prime(3)=5 because it is the 5th prime. There are 3 twin prime subsets on the set {5,6,7,...,24,25} so the 3rd term is 3.
%t A273257 Table[Function[w, Length@ Select[Prime[Range @@ w], Function[p, And[# - p == 2, # < Prime@ Last@ w] &@ NextPrime@ p]]]@ {n, PrimePi[Prime[n]^2]}, {n, 55}] (* _Michael De Vlieger_, Aug 30 2016 *)
%t A273257 ntp[n_]:=Count[Partition[Select[Range[Prime[n],Prime[n]^2],PrimeQ],2,1], _?(#[[2]]-#[[1]]==2&)]; Join[{0,1},Array[ntp,60,3]] (* _Harvey P. Dale_, Nov 01 2016 *)
%o A273257 (PARI) a(n)=if(n<3,return(n-1)); my(p=prime(n),q=p,s); forprime(r=q+1,p^2, if(r-q==2, s++); q=r); s \\ _Charles R Greathouse IV_, Aug 28 2016
%Y A273257 Cf. A001097, A079047, A143738.
%K A273257 nonn
%O A273257 1,3
%A A273257 _Jesse H. Crotts_, Aug 28 2016
%E A273257 More terms from _Charles R Greathouse IV_, Aug 28 2016