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

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

Author

Antti Karttunen and Labos Elemer, Jun 01 2004

Keywords

• nonn

Comments

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

Links

• 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)]

Crossrefs

Cf. A033843, A095016, A036378.

Programs

Mathematica

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 *)

PARI

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

Extensions

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