OFFSET
1,1
COMMENTS
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)?
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
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000
J. B. Rosser, The n-th Prime is Greater than n log(n), Proc. London Math. Soc. 45, 21-44, 1939.
Eric Weisstein's World of Mathematics, Semiprime.
FORMULA
a(n) ~ n log^2 n / log log n. - Charles R Greathouse IV, Dec 28 2011
EXAMPLE
a(1) = semiprime(prime(1)) = semiprime(2) = 6.
a(2) = semiprime(prime(2)) = semiprime(3) = 9.
MAPLE
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
MATHEMATICA
terms = 55;
semiPrimes = Select[Range[16 terms], PrimeOmega[#] == 2&];
(* 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. *)
a[n_] := semiPrimes[[Prime[n]]];
Array[a, terms] (* Jean-François Alcover, Apr 13 2020 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Apr 30 2005
EXTENSIONS
All values after a(32) corrected by R. J. Mathar, Dec 14 2009
STATUS
approved