This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A171727 #42 Apr 24 2021 16:38:59
%S A171727 1,1,1,1,2,2,4,1,3,2,2,4,7,3,3,5,7,4,4,7,6,11,9,5,11,9,9,11,10,11,9,
%T A171727 11,11,12,11,12,18,12,12,16,11,16,20,14,16,15,20,16,22,13,22,16,17,21,
%U A171727 20,20,23,22,23,20,21,21,26,20,28,24,24,23,24,25,21,24,37,27,21,28,24,31
%N A171727 The number of twin prime pairs in the interval (p^2,p*q), where (p,q) runs over the twin prime pairs (A001359(n),A006512(n)).
%C A171727 If you graph the order of the twin primes along the x-axis (i.e., first twin, second, third, ...) and the number of twins in the sequence given above along the y-axis, a clear pattern emerges. As you go farther along the x-axis, the number of twin primes, on average, within the interval increases. The pattern appears to be nonlinear. If one could prove that there's at least one twin prime within each interval, the twin prime conjecture would be proved since the n-th twin produces larger intervals with more twin primes. The evidence seems overwhelming.
%D A171727 C. C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Perseus Books, 1999.
%D A171727 J. Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Penguin Books Canada Ltd., 2004.
%D A171727 M. du Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, HarperCollins Publishers Inc., 2004.
%H A171727 J. S. Cheema, <a href="/A171727/b171727.txt">Table of n, a(n) for n = 1..1044</a>
%e A171727 The first twin prime pair (3,5) corresponds to the interval (9,15), which contains one twin prime pair (11,13), so a(1) = 1.
%e A171727 The fifth twin prime pair (29,31) corresponds to the interval (841,899), which contains the twin prime pairs (857,859) and (881,883), so a(5) = 2.
%o A171727 (PARI) {for(k=1, 300, if(prime(k+1)-prime(k)==2, my(c=0); forprime(m=prime(k)^2, prime(k)*prime(k+1), c+=isprime(m+2)); print1(c, ", ")))} \\ _Zhandos Mambetaliyev_, Mar 28 2021
%Y A171727 Cf. A001359, A006512, A108570, A037074. - _Charlie Neder_, Feb 12 2019
%K A171727 nonn
%O A171727 1,5
%A A171727 _Jaspal Singh Cheema_, Dec 16 2009
%E A171727 Partially edited by _Michel Marcus_, Mar 19 2013
%E A171727 Edited by _Charlie Neder_, Feb 12 2019