A106350 Semiprimes indexed by primes.

6, 9, 14, 21, 33, 35, 49, 55, 65, 86, 91, 115, 122, 129, 142, 159, 183, 187, 206, 215, 218, 247, 259, 287, 303, 319, 323, 334, 339, 358, 403, 415, 446, 451, 482, 489, 511, 527, 537, 553, 573, 581, 626, 633, 655, 667, 698, 737, 753, 758, 771, 791, 794, 835, 851

Offset: 1

Author

Jonathan Vos Post, Apr 30 2005

Keywords

• easy
• nonn

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

Examples

			a(1) = semiprime(prime(1)) = semiprime(2) = 6.
a(2) = semiprime(prime(2)) = semiprime(3) = 9.
		

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.

Crossrefs

Cf. A000040, A001358, A007097, A091022, A105997, A105998, A106349.

Programs

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 *)

Formula

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

a(n) ~ n log^2 n / log log n. - Charles R Greathouse IV, Dec 28 2011

Extensions

All values after a(32) corrected by R. J. Mathar, Dec 14 2009