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A106350 Semiprimes indexed by primes. 16
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 (list; graph; refs; listen; history; text; internal format)
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

REFERENCES

Rosser, J. B. "The n-th Prime is Greater than n log(n)." Proc. London Math. Soc. 45, 21-44, 1938.

LINKS

Table of n, a(n) for n=1..55.

Eric Weisstein's World of Mathematics, Semiprime.

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

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

CROSSREFS

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

Sequence in context: A316013 A154778 A316014 * A217851 A218697 A316015

Adjacent sequences:  A106347 A106348 A106349 * A106351 A106352 A106353

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

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Last modified October 23 12:45 EDT 2019. Contains 328345 sequences. (Running on oeis4.)