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A066265
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a(n) = number of semiprimes < 10^n.
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27
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0, 3, 34, 299, 2625, 23378, 210035, 1904324, 17427258, 160788536, 1493776443, 13959990342, 131126017178, 1237088048653, 11715902308080, 111329817298881, 1061057292827269, 10139482913717352, 97123037685177087, 932300026230174178, 8966605849641219022, 86389956293761485464
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OFFSET
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0,2
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COMMENTS
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LINKS
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Eric Weisstein's World of Mathematics, Semiprime
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FORMULA
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(1/2)*( pi(10^(n/2)) + Sum_{i=1..pi(10^n)} pi( (10^n-1)/P_i) ) = Sum_{i=1..pi(sqrt(10^n))} pi( (10^n-1)/P_i ) - binomial( pi(sqrt(10^n)), 2). - Robert G. Wilson v, May 16 2005
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EXAMPLE
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Below 10 there are three semiprimes: 4 (2*2), 6 (2*3) and 9 (3*3).
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MATHEMATICA
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f[n_] := Sum[ PrimePi[(10^n - 1)/Prime[i]], {i, PrimePi[ Sqrt[10^n]]}] - Binomial[ PrimePi[ Sqrt[10^n]], 2]; Do[ Print[ f[n]], {n, 0, 14}] (* Robert G. Wilson v, May 16 2005 *)
SemiPrimePi[n_] := Sum[ PrimePi[n/Prime@ i] - i + 1, {i, PrimePi@ Sqrt@ n}]; Array[ SemiPrimePi[10^# - 1] &, 14, 0] (* Robert G. Wilson v, Jan 21 2015 *)
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PROG
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(PARI) a(n)=my(s); forprime(p=2, sqrt(10^n), s+=primepi((10^n-1)\p)); s-binomial(primepi(sqrt(10^n)), 2) \\ Charles R Greathouse IV, Apr 23 2012
(Perl) use Math::Prime::Util qw/:all/; use integer; sub countsp { my($k, $sum, $pc)=($_[0]-1, 0, 1); prime_precalc(60_000_000); forprimes { $sum += prime_count($k/$_) + 1 - $pc++; } int(sqrt($k)); $sum; } foreach my $n (0..16) { say "$n: ", countsp(10**$n); } # Dana Jacobsen, May 11 2014
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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