A129542 Number of isolated primes < 10^n.

1, 10, 99, 820, 7145, 62161, 546620, 4880832, 43998523, 400227154, 3669302718, 33866741579, 314396207096, 2933381107473, 27490151938062, 258629969639330, 2441659478947916, 23122602510585989

Offset: 1

Author

Cino Hilliard, Jun 08 2007

Keywords

• hard
• more
• nonn

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.

Examples

			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.
		

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]

Crossrefs

Cf. A006880, A007508.

Programs

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",") ) }

Formula

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

Extensions

Edited by Max Alekseyev, Apr 27 2009