A111067 Number of odd primes p < 10^n such that p+2 = product of 2 primes (no twin Chen primes).

1, 11, 79, 427, 3009, 21779, 166649, 1322266, 10752066, 89305602, 754868608, 6472917998

Offset: 1

Author

Pierre CAMI, Oct 08 2005

Keywords

• nonn
• hard
• more

Comments

A006880(n) = number of primes < 10^n, A007508(n) = number of twin primes < 10^n. Let F(n) = A006880(n)/A007508(n). For n > 3, we find that F(n) is ~ 0.762373*log(10^n) - 0.968855.

Let FF(n) = A006880(n)/a(n). For n>3, we find that FF(n) is ~ 0.163128*log(10^n) + 1.349255. a(n)/A007508(n) is then ~ 0.762373*log((10^n) - 0.968855 / ( 0.163128*log(10^n) + 1.349255, as n tends to infinity a(n) / A007508(n) needs to tend to 0.762373 / 0.163128 = 4.673465.

Examples

			7 is the only prime < 10 with 7+2 = 3*3 = product of 2 odd primes so a(1) = 1.
		

Crossrefs

Cf. A006880, A007508, A063637.

Programs

Mathematica

a[n_] := Count[Prime[Range[2, PrimePi[10^n]]], ?(PrimeOmega[# + 2] == 2 &)]; Array[a, 6] (* _Amiram Eldar, Jul 25 2025 *)

PARI

list(len) = {my(c = 0, k = 0, pow = 10); forprime(p = 3, , if(p > pow, print1(c, ", "); k++; if(k == len, break); pow *= 10); if(bigomega(p+2) == 2, c++));} \\ Amiram Eldar, Jul 25 2025

Extensions

a(8) corrected and a(9) computed by Robert G. Wilson v, Oct 10 2005

a(10)-a(12) from Amiram Eldar, Jul 25 2025