A033843 -id:A033843 - cp's OEIS frontend

A071538 Number of twin prime pairs (p, p+2) with p <= n.

0, 0, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 1

Reinhard Zumkeller, May 30 2002

nonn

The convention is followed that a twin prime is <= n if its smaller member is <= n.
Except for (3, 5), every pair of twin primes is congruent (-1, +1) (mod 6). - Daniel Forgues, Aug 05 2009
This function is sometimes known as pi_2(n). If this name is used, there is no obvious generalization for pi_k(n) for k > 2. - Franklin T. Adams-Watters, Jun 01 2014

			a(30) = 5, since (29,31) is included along with (3,5), (5,7), (11,13) and (17,19).

S. Lang, The Beauty of Doing Mathematics, pp. 12-15; 21-22, Springer-Verlag NY 1985.

Daniel Forgues, Table of n, a(n) for n = 1..99998
Thomas R. Nicely, Some Results of Computational Research in Prime Numbers.
Eric Weisstein's World of Mathematics, Twin Primes.

Cf. A007508, A033843, A001359, A006512.

primePi2[1] = 0; primePi2[n_] := primePi2[n] = primePi2[n - 1] + Boole[PrimeQ[n] && PrimeQ[n + 2]]; Table[primePi2[n], {n, 100}] (* T. D. Noe, May 23 2013 *)

A071538(n) = local(s=0,L=0); forprime(p=3,n+2,L==p-2 & s++; L=p); s
/* For n > primelimit, one may use: */ A071538(n) = { local(s=isprime(2+n=precprime(n))&n,L); while( n=precprime(L=n-2),L==n & s++); s }
/* The following gives a reasonably good estimate for small and for large values of n (cf. A007508): */
A071538est(n) = 1.320323631693739*intnum(t=2,n+1/n,1/log(t)^2)-log(n) /* (The constant 1.320... is A114907.) */ \\ M. F. Hasler, Dec 10 2008

Definition edited by Daniel Forgues, Jul 29 2009

A095017 Number of lesser twin primes (A001359) in range ]2^n, 2^(n+1)].

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 7, 7, 12, 26, 45, 70, 113, 215, 355, 666, 1153, 2071, 3785, 6965, 12495, 22643, 41608, 76371, 140944, 261752, 484968, 904799, 1689477, 3160113, 5928904, 11139071, 20970782, 39535081, 74697745, 141342490, 267812262, 508194094, 965623233, 1837147717

Offset: 1

Antti Karttunen and Labos Elemer, Jun 01 2004

nonn

Conjecture: a(n) > 0 for all n. This holds for all n <= 100. - Charles R Greathouse IV, May 14 2012

It appears that a(n+1) is approximately 2 * a(n) * (n/(n+1))^2. The 2 accounts for each segment of numbers being twice as large as the previous segment. The first n/(n+1) accounts for primes being less common as numbers increase in size. The second n/(n+1) accounts for twin primes being a less common gap size as numbers increase in size. This formula has increasing accuracy as the numbers increase and is better than 0.1% by the end of the known sequence. - Jerry M Lagrou, Jan 05 2025

Jerry M Lagrou, Table of n, a(n) for n = 1..52
Igoris Belovas and Martynas Sabaliauskas, On integer sequences associated with admissible prime k-tuples, Ann. Univ. Craiova-Math. Comp. Sci. Series (2025) Vol. 52, No. 1.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

Cf. A033843, A095016, A036378.

Table[ps = Prime[Range[PrimePi[2^n] + 1, PrimePi[2^(n+1) + 1]]]; Count[Differences[ps], 2], {n, 25}] (* T. D. Noe, May 08 2012 *)

a095017(maxex2)={my (L=List([1]), p2=8, n2=0, pp=5); forprime (p=7, 2^maxex2, if (p>p2, p2*=2; listput(L,n2); n2=0); if (p-pp==2, n2++); pp=p); Vec(L)};
a095017(30) \\ Hugo Pfoertner, Feb 05 2024

a(34) and beyond from Jerry M Lagrou, Dec 02 2023

A215473 Number of prime quadruples with smallest member < 2^n.

Original entry on oeis.org

0, 0, 1, 2, 2, 2, 3, 4, 4, 5, 7, 10, 11, 16, 23, 28, 43, 62, 106, 177, 309, 483, 795, 1305, 2105, 3525, 5923, 10096, 17259, 30004

Offset: 1

Alex Ratushnyak, Aug 12 2012

nonn

Prime quadruples (A007530) are numbers n such that n, n+2, n+6, n+8 are all prime.

			a(3) = 1 because there is only one prime quadruple below 2^3, namely {5, 7, 11, 13}.
a(4) = 2 because there are two prime quadruples below 2^4: the aforementioned and {11, 13, 17, 19}.

Cf. A050258, similar definition but with powers of 10 instead of 2.
Cf. A007530, A014561, A112540.
Cf. A033843, A007053.

(* First run program for A007530 *) Table[Length[Select[A007530, # < 2^n &]], {n, 14}] (* Alonso del Arte, Aug 12 2012 *)

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A071538 Number of twin prime pairs (p, p+2) with p <= n.

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A095017 Number of lesser twin primes (A001359) in range ]2^n, 2^(n+1)].

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Mathematica

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A215473 Number of prime quadruples with smallest member < 2^n.

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