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A106350 Semiprimes indexed by primes.

%I A106350 #26 Feb 16 2025 08:32:57
%S A106350 6,9,14,21,33,35,49,55,65,86,91,115,122,129,142,159,183,187,206,215,
%T A106350 218,247,259,287,303,319,323,334,339,358,403,415,446,451,482,489,511,
%U A106350 527,537,553,573,581,626,633,655,667,698,737,753,758,771,791,794,835,851
%N A106350 Semiprimes indexed by primes.
%C A106350 This is the sequence of the n-th semiprime for n = {2,3,5,7,11,13,17,19,23,29...}. Not to be confused with A106349: Primes indexed by semiprimes. We seek to know what this sequence is asymptotically, as J. B. Rosser's result, subsequently modified, is that prime(n) ~ n*(log n + log log n - 1). hence semiprime(prime(n)) ~ semiprime(n)*(log semiprime(n) + log log semiprime(n) - 1). But what is, asymptotically, semiprime(n)?
%C A106350 Semiprime(n) ~ n log n / log log n, hence a(n) ~ n log^2 n / log log n. - _Charles R Greathouse IV_, Dec 28 2011
%H A106350 Alois P. Heinz, <a href="/A106350/b106350.txt">Table of n, a(n) for n = 1..10000</a>
%H A106350 J. B. Rosser, <a href="https://doi.org/10.1112/plms/s2-45.1.21">The n-th Prime is Greater than n log(n)</a>, Proc. London Math. Soc. 45, 21-44, 1939.
%H A106350 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Semiprime.html">Semiprime.</a>
%F A106350 a(n) = semiprime(prime(n)). a(n) = A001358(A000040(n)).
%F A106350 a(n) ~ n log^2 n / log log n. - _Charles R Greathouse IV_, Dec 28 2011
%e A106350 a(1) = semiprime(prime(1)) = semiprime(2) = 6.
%e A106350 a(2) = semiprime(prime(2)) = semiprime(3) = 9.
%p A106350 A001358 := proc(n) if n = 1 then 4; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 2 then return a ; end if; end do ; end if ; end proc: A106350 := proc(n) A001358(ithprime(n)) ; end proc: seq(A106350(n),n=1..80) ; # _R. J. Mathar_, Dec 14 2009
%t A106350 terms = 55;
%t A106350 semiPrimes = Select[Range[16 terms], PrimeOmega[#] == 2&];
%t A106350 (* NB If the index Prime[terms] exceeds the size of the table semiPrimes, then the coefficient 16 has to be increased according to the number of terms desired: for instance, for 1000 terms, replace 16 with 32. *)
%t A106350 a[n_] := semiPrimes[[Prime[n]]];
%t A106350 Array[a, terms] (* _Jean-François Alcover_, Apr 13 2020 *)
%Y A106350 Cf. A000040, A001358, A007097, A091022, A105997, A105998, A106349.
%K A106350 easy,nonn
%O A106350 1,1
%A A106350 _Jonathan Vos Post_, Apr 30 2005
%E A106350 All values after a(32) corrected by _R. J. Mathar_, Dec 14 2009