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%I #42 Apr 24 2021 16:38:59
%S 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 11,11,12,11,12,18,12,12,16,11,16,20,14,16,15,20,16,22,13,22,16,17,21,
%U 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 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 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 C. C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Perseus Books, 1999.
%D J. Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Penguin Books Canada Ltd., 2004.
%D M. du Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, HarperCollins Publishers Inc., 2004.
%H J. S. Cheema, <a href="/A171727/b171727.txt">Table of n, a(n) for n = 1..1044</a>
%e 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 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 (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 Cf. A001359, A006512, A108570, A037074. - _Charlie Neder_, Feb 12 2019
%K nonn
%O 1,5
%A _Jaspal Singh Cheema_, Dec 16 2009
%E Partially edited by _Michel Marcus_, Mar 19 2013
%E Edited by _Charlie Neder_, Feb 12 2019
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