A088018 -id:A088018 - cp's OEIS frontend

A088019 Number of twin primes between n and 2n (inclusive).

0, 1, 2, 2, 2, 2, 3, 2, 3, 4, 4, 3, 3, 2, 3, 4, 4, 3, 3, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6, 5, 5, 4, 4, 4, 4, 4, 4, 4, 5, 6, 6, 7, 8, 8, 8, 8, 8, 7, 7, 6, 6, 6, 6, 6, 6, 6, 7, 8, 8, 7, 7, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 10, 10, 10, 11, 12, 12, 13, 14

Offset: 1

T. D. Noe, Sep 18 2003

Here a twin prime is counted even if only one member of the twin-prime pair is between n and 2n, inclusive. Note that this sequence is very close to 2*A088018. It appears that a(n) > 0 for all n > 1. However, it has not been proved that there are an infinite number of twin primes.

T. D. Noe, Table of n, a(n) for n=1..10000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A035250 (number of primes between n and 2n), A088018 (number of twin-prime pairs between n and 2n).

pl=Prime[Range[PrimePi[20000]]]; twl={}; Do[If[pl[[i-1]]+2==pl[[i]], twl=Join[twl, {pl[[i-1]], pl[[i]]}]], {i, 2, Length[pl]}]; twl=Union[twl]; i1=1; i2=1; nMin=(twl[[1]]-1)/2; nMax=(twl[[ -1]]+1)/2; Join[Table[0, {nMin-1}], Table[While[twl[[i1]]

A186312 There appear to be at least n lower twin primes between x/2 and x for all x >= a(n).

Original entry on oeis.org

11, 59, 101, 149, 179, 227, 569, 599, 641, 809, 821, 1019, 1049, 1061, 1289, 1319, 1427, 1451, 1481, 1667, 1787, 1871, 1877, 1931, 1949, 2081, 2129, 2237, 2657, 2687, 2711, 2999, 3251, 3257, 3299, 3359, 3461, 3467, 3527, 3539, 3767, 3917, 4001, 4019, 4091

Offset: 1

T. D. Noe, Feb 17 2011

nonn

The name says "appear" because this sequence assumes that the twin prime conjecture is true. Note that every term is also a lower twin prime. This sequence for lower twin primes is like the Ramanujan primes (A104272) for the sequence of prime numbers. However, in this case, we do not know if there are an infinite number of twin primes. It appears that A001359(2n) < a(n) < A001359(4n).

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001359 (lower twin primes), A088018 (counts the lower twin primes between n-1 and 2(n-1)).

cp's OEIS Frontend

A088019 Number of twin primes between n and 2n (inclusive).

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