1,2

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.

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.

a(n) = A006880(n) - 2*A007508(n) + 1

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.

(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", ") ) }

Cf. A006880, A007508.

Sequence in context: A163446 A007137 A135927 this_sequence A081109 A004189 A105694

Adjacent sequences: A129539 A129540 A129541 this_sequence A129543 A129544 A129545

hard,more,nonn

Cino Hilliard (hillcino368(AT)hotmail.com), Jun 08 2007

Edited by Max Alekseyev (maxale(AT)gmail.com), Apr 27 2009