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A171727
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The number of twin primes within a given interval.
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1
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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, 11, 11, 12, 11, 12, 18, 12, 12, 16, 11, 16, 20, 14, 16, 15, 20, 16, 22, 13, 22, 16, 17, 21, 20, 20, 23, 22, 23, 20, 21, 21, 26, 20, 28, 24, 24, 23, 24, 25, 21, 24, 37, 27, 21, 28, 24, 31
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OFFSET
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1,5
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COMMENTS
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The given interval is defined as [A001359(n)^2, A001359(n)*(A001359(n)+2)].
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, greater are the number of twin primes, on average, within the interval obtained. The pattern appears to be nonlinear. If one can 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.
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REFERENCES
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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. D. Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, HarperCollins Publishers Inc., 2004.
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LINKS
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J. S. Cheema, Table of n, a(n) for n=1,...,1044.
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EXAMPLE
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Take any pair of twin primes. Let us say the very first one (3,5). Square the first term, you get 9, then take the product of the two primes, you get 15. Between these two numbers, namely (9,15) there is 1 twin prime (11,13). Hence the very first term of the sequence is 1.
As another example, take the fifth twin prime pair (29,31) - (the initial terms of the first five are 3,5,11,17,29). Square the first term and then take the product of the two primes to obtain an interval (841,899). Between these two numbers, there are 2 twin primes (857,859) and (881,883). Hence the fifth term in the sequence above is 2, and so on.
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CROSSREFS
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Sequence in context: A212791 A175001 A205843 * A171942 A214714 A023137
Adjacent sequences: A171724 A171725 A171726 * A171728 A171729 A171730
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KEYWORD
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nonn,uned
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AUTHOR
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Jaspal Singh Cheema (Jaspal(AT)rogers.com), Dec 16 2009
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EXTENSIONS
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Partially edited by Michel Marcus, Mar 19 2013
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STATUS
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approved
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