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

%I A095017 #46 Jul 08 2025 14:47:44
%S A095017 1,1,1,2,2,3,7,7,12,26,45,70,113,215,355,666,1153,2071,3785,6965,
%T A095017 12495,22643,41608,76371,140944,261752,484968,904799,1689477,3160113,
%U A095017 5928904,11139071,20970782,39535081,74697745,141342490,267812262,508194094,965623233,1837147717
%N A095017 Number of lesser twin primes (A001359) in range ]2^n, 2^(n+1)].
%C A095017 Conjecture: a(n) > 0 for all n. This holds for all n <= 100. - _Charles R Greathouse IV_, May 14 2012
%C A095017 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
%H A095017 Jerry M Lagrou, <a href="/A095017/b095017.txt">Table of n, a(n) for n = 1..52</a>
%H A095017 Igoris Belovas and Martynas Sabaliauskas, <a href="https://doi.org/10.52846/ami.v52i1.1988">On integer sequences associated with admissible prime k-tuples</a>, Ann. Univ. Craiova-Math. Comp. Sci. Series (2025) Vol. 52, No. 1.
%H A095017 <a href="/index/Pri#primesubsetpop2">Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]</a>
%t A095017 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 *)
%o A095017 (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)};
%o A095017 a095017(30) \\ _Hugo Pfoertner_, Feb 05 2024
%Y A095017 Cf. A033843, A095016, A036378.
%K A095017 nonn
%O A095017 1,4
%A A095017 _Antti Karttunen_ and _Labos Elemer_, Jun 01 2004
%E A095017 a(34) and beyond from _Jerry M Lagrou_, Dec 02 2023