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A114021 Number of semiprimes between n and n + sqrt(n). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

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

LINKS

T. D. Noe, Table of n, a(n) for n = 0..10000

FORMULA

a(n) = card{S such that S is an element of A001358 and n < S < n + n^(1/2)}.

EXAMPLE

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...

MATHEMATICA

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 *)

PROG

(Perl) use ntheory ":all"; print "$_ ", semiprime_count($_+1, $_+sqrtint($_)-($_ && is_square($_))), "\n" for 0..1000; # Dana Jacobsen, Mar 04 2019

CROSSREFS

Cf. A001358, A060715, A065516, A077463, A085809, A097824, A114412, A115766.

Sequence in context: A303903 A209318 A170984 * A239287 A305258 A053616

Adjacent sequences:  A114018 A114019 A114020 * A114022 A114023 A114024

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Jan 31 2006

EXTENSIONS

Corrected and extended by T. D. Noe, Aug 12 2008

STATUS

approved

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Last modified April 25 18:12 EDT 2019. Contains 322461 sequences. (Running on oeis4.)