A106349 Primes indexed by semiprimes.

7, 13, 23, 29, 43, 47, 73, 79, 97, 101, 137, 139, 149, 163, 167, 199, 227, 233, 257, 269, 271, 293, 313, 347, 373, 389, 421, 439, 443, 449, 467, 487, 491, 499, 577, 607, 631, 647, 653, 661, 673, 677, 727, 751, 757, 811, 821, 823, 829, 839, 907, 929, 937, 947

Offset: 1

Author

Jonathan Vos Post, Apr 29 2005

Keywords

• nonn
• easy

Comments

This is the sequence of the k-th prime for k = {4,6,9,10,14,15,21,22,25,26,33,34,35,38,39,46,49,51,...}. Not to be confused with A106350: semiprimes indexed by primes.

Examples

			a(1) = 7 because semiprime(1) = 4, so prime(semiprime(1)) = prime(4) = 7.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Paul Kinlaw, Megan Triplett, and William Tripp, Almost Primes of Almost Prime Index, INTEGERS, Vol 24 (2024), Article #A99.

Crossrefs

Cf. A000040, A000720, A001358, A007097, A091022, A105997, A105998, A106350.

Programs

Magma

[NthPrime(n): n in [2..200] | &+[d[2]: d in Factorization(n)] eq 2]; // Vincenzo Librandi, Nov 28 2015

Mathematica

Prime@ Select[Range@ 161, PrimeOmega@ # == 2 &] (* or *) Select[Prime@ Range@ 161, PrimeOmega@ PrimePi@ # == 2 &] (* Michael De Vlieger, Nov 28 2015 *)

PARI

lista(nn) = select(x->(bigomega(primepi(x))==2), primes(nn)); \\ Michel Marcus, Nov 29 2015

Formula

a(n) = prime(semiprime(n)).

a(n) = A000040(A001358(n)).

pi(a(n)) = p*q for some primes p and q.

Sum_{n>=1} 1/a(n) is in the interval (0.9910, 0.9915) (Kinlaw et al., 2024, Theorem 6, p. 11). - Amiram Eldar, Nov 09 2024