OFFSET
1,5
COMMENTS
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.
REFERENCES
C. C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Perseus Books, 1999.
J. Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Penguin Books Canada Ltd., 2004.
M. du Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, HarperCollins Publishers Inc., 2004.
LINKS
J. S. Cheema, Table of n, a(n) for n = 1..1044
EXAMPLE
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.
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.
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Jaspal Singh Cheema, Dec 16 2009
EXTENSIONS
Partially edited by Michel Marcus, Mar 19 2013
Edited by Charlie Neder, Feb 12 2019
STATUS
approved