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

0, 1, 3, 4, 8, 9, 16, 17, 21, 29, 30, 41, 48, 50, 61, 74, 87, 91, 110, 121, 123, 138, 152, 166, 187, 202, 208, 218, 223, 234, 276, 288, 315, 320, 365, 374, 394, 411, 432, 455, 480, 492, 541, 547, 567, 574, 626, 685, 708, 716, 732, 764, 772, 818, 851

Offset

1,3

Comments

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).

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.

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

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Example

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.

Mathematica

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 *)
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 *)

Prog

(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

Crossrefs

Cf. A001097, A079047, A143738.

Sequence in context: A123722 A328733 A217788 * A249485 A254877 A193351

Adjacent sequences: A273254 A273255 A273256 * A273258 A273259 A273260

Keyword

nonn

Author

Jesse H. Crotts, Aug 28 2016

Extensions

More terms from Charles R Greathouse IV, Aug 28 2016

Status

approved