

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



