OFFSET
1,2
COMMENTS
Isolated primes are primes that are not twin prime components. Define I(n) to be the number of isolated primes <= n. Given that Pi(n) -> infinity and I(n) -> infinity as n -> infinity, proving that pi(n) always grows by an ever so slight factor k>1 than I(n), then we will have infinity_Pi(n) - infinity_I(n) = infinity. So twin primes would be infinite in extent.
LINKS
C. Hilliard, Sum Isolated Primes.
C. Hilliard, Gcc code. It took 7.5 hrs to compute a(12). It will take the Gcc program 3.2 days to compute a(13). For a(16) it will take about 8 years.
Cino Hilliard, Sum of Isolated primes, message 38 in seqfun Yahoo group, providing code for gcc (needs formatting to become compilable), Jun 5, 2007. [Cached copy]
EXAMPLE
The 10 isolated primes < 10^2 are 2,23,37,47,53,67,79,83,89,97 so 10 is the second entry in the table.
PROG
(PARI) countisoprimes(n) = \Count primes that are not twin prime components < 10^n { local(j, c, x); for(j=1, n, c=0; forprime(x=2, 10^j, if(!isprime(x-2)&&!isprime(x+2), c++) ); print1(c", ") ) }
CROSSREFS
KEYWORD
hard,more,nonn
AUTHOR
Cino Hilliard, Jun 08 2007
EXTENSIONS
Edited by Max Alekseyev, Apr 27 2009
STATUS
approved