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

1,2

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

Table of n, a(n) for n=1..18.

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]

Formula

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

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

Cf. A006880, A007508.

Sequence in context: A135927 A299952 A278672 * A359482 A224752 A224761

Adjacent sequences: A129539 A129540 A129541 * A129543 A129544 A129545

Keyword

hard,more,nonn

Author

Cino Hilliard, Jun 08 2007

Extensions

Edited by Max Alekseyev, Apr 27 2009

Status

approved