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A114021
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Number of semiprimes between n and n + sqrt(n).
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4
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0, 0, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 2, 2, 2, 2, 2, 1, 0, 0, 1, 2, 3, 3, 3, 3, 3, 2, 2, 2, 1, 0, 1, 1, 1, 2, 2, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 5, 5, 5, 4, 4, 4, 4, 3, 3, 2, 1, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2
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OFFSET
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0,9
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COMMENTS
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It appears that for n > 37 it is always true that a(n) > 0. The exponent can be reduced further. Since 597 + 597^(0.4129) > 611, leaping the record semiprime gap between 597 and 611, it seems that for n > 597 it is always true that there is a semiprime between n and n^(0.4129). It seems that for n > 2705 it is always true that there is a semiprime between n and n^(0.3509). These conjectures are related to the various sequences about semiprime gaps and the merit of such gaps.
a(96) appears to be the last zero term. - T. D. Noe, Aug 12 2008
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LINKS
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FORMULA
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a(n) = card{S such that S is an element of A001358 and n < S < n + n^(1/2)}.
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EXAMPLE
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a(0) = 0 because there are no semiprimes between 0 and 0+sqrt(0) = 0.
a(2) = 0 because there are no semiprimes between 2 and 2+sqrt(2) = 3.414...
a(3) = 1 as the semiprime 4 falls between 3 and 3 + sqrt(3) = 4.732...
a(5) = 1 as the semiprime 6 falls between 5 and 5 + sqrt(5) = 7.236...
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MATHEMATICA
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SemiPrimeQ[n_] := TrueQ[Plus@@Last/@FactorInteger[n]==2]; Table[hi=n+Sqrt[n]; If[IntegerQ[hi], hi--, hi=Floor[hi]]; Length[Select[Range[n+1, hi], SemiPrimeQ]], {n, 0, 150}] (* T. D. Noe, Aug 12 2008 *)
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PROG
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(Perl) use ntheory ":all"; print "$_ ", semiprime_count($_+1, $_+sqrtint($_)-($_ && is_square($_))), "\n" for 0..1000; # Dana Jacobsen, Mar 04 2019
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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Corrected and extended by T. D. Noe, Aug 12 2008
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STATUS
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approved
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