|
%I #34 Jul 04 2019 03:42:50
%S 0,1,3,4,8,9,16,17,21,29,30,41,48,50,61,74,87,91,110,121,123,138,152,
%T 166,187,202,208,218,223,234,276,288,315,320,365,374,394,411,432,455,
%U 480,492,541,547,567,574,626,685,708,716,732,764,772,818,851
%N Number of twin primes between prime(n) and prime(n)^2.
%C 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 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 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 Charles R Greathouse IV, <a href="/A273257/b273257.txt">Table of n, a(n) for n = 1..10000</a>
%e 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 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 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 (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 Cf. A001097, A079047, A143738.
%K nonn
%O 1,3
%A _Jesse H. Crotts_, Aug 28 2016
%E More terms from _Charles R Greathouse IV_, Aug 28 2016
|
