Re: How to reverse sieve for prime numbers?
"Martin DeMello [email protected] [ocaml_beginners]" <[email protected]> Thu, 16 Feb 2017 00:49:02 -0800
| Newsgroups | gmane.comp.lang.ocaml.beginners |
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| Message-ID | <CAFrFfuG6QiYm6WUSxTgJqxe2_rN6Ei+owV3+QDnJGv1KVjSe1g@mail.gmail.com> |
--001a1142dbde7378d60548a1deac Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable This is a really good comparison between ocaml and sml: http://adam.chlipala.net/mlcomp/ As for finding the first n primes, that requires a bit of mathematics. Starting with the prime counting function [ https://en.wikipedia.org/wiki/Prime-counting_function], it can be shown that the nth prime is less than n * [ln(n) + ln(ln(n))] where ln is the natural log function and n > 6. So what you need to do is calculate the upper bound, run your prime sieve, and then take the first n primes from there. martin On Wed, Feb 15, 2017 at 11:59 PM, Douglas Lewit [email protected] [ocaml_beginners] <[email protected]> wrote: > > > Hi everyone, > > This is a function I created to filter out all the prime numbers from 2 t= o > some integer n. I like the function and it appears to be very fast and > efficient. > > *let sieve num =3D let rec sequence lower upper =3D if lower > upper then= [ ] > else lower :: sequence ( lower + 1 ) upper in * > * let integers =3D sequence 2 num in * > * let num' =3D float_of_int num in * > * let num'' =3D sqrt num' in * > * let num''' =3D ceil num'' in * > * let num'''' =3D int_of_float num''' in * > * let rec sieve' start potential_primes =3D * > * if start > num'''' then potential_primes= * > * else sieve' ( start + 1 ) ( List.filter = ( > fun x -> if x > start && x mod start =3D 0 then false else true ) * > * potential_primes ) in * > * sieve' 2 integers ;;* > > > However, I'm having trouble with the reverse question. I know from > experimentation that there are 25 prime integers in the interval [2, 100)= . > That's great. I can find those integers by apply the function above with= *sieve > 100*. But what a function called *findNPrimes 25* that returns the first > 25 prime numbers? I'm finding that function rather difficult to write. > Any clues or hints? Thanks!!! > > Best, > > Douglas. > > P.S. Does anyone know of any CS professors or professionals in the > Chicago area who are fairly adept at OCaml? Also, I am curious about > something. I read that OCaml really descended from a programming languag= e > called STANDARD ML. How different is OCaml from Standard ML. Pretty > close? Or very different? I appreciate the feedback. > > > > >=20 > --001a1142dbde7378d60548a1deac Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN" "http://www.w3.org/TR/htm= l4/strict.dtd"> <html> <head> </head> =20 <body style=3D"background-color: #fff;"> <span style=3D"display:none"> </span> <!--~-|**|PrettyHtmlStartT|**|-~--> <div id=3D"ygrp-mlmsg" style=3D"position:relative;"> <div id=3D"ygrp-msg" style=3D"z-index: 1;"> <!--~-|**|PrettyHtmlEndT|**|-~--> <div id=3D"ygrp-text" > =20=20=20=20=20=20 =20=20=20=20=20=20 <p><div dir=3D"ltr">This is a really good comparison between ocaml an= d sml:<div><br></div><div><a href=3D"http://adam.chlipala.net/mlcomp/">http= ://adam.chlipala.net/mlcomp/</a><br></div><div><br></div><div>As for findin= g the first n primes, that requires a bit of mathematics. Starting with the= prime counting function [<a href=3D"https://en.wikipedia.org/wiki/Prime-co= unting_function">https://en.wikipedia.org/wiki/Prime-counting_function</a>]= , it can be shown that the nth prime is less than n * [ln(n) + ln(ln(n))] w= here ln is the natural log function and n > 6. So what you need to do is= calculate the upper bound, run your prime sieve, and then take the first n= primes from there.</div><div><br></div><div>martin</div></div><div class= =3D"gmail_extra"><br><div class=3D"gmail_quote">On Wed, Feb 15, 2017 at 11:= 59 PM, Douglas Lewit <a href=3D"mailto:[email protected]">[email protected]= </a> [ocaml_beginners] <span dir=3D"ltr"><<a href=3D"mailto:ocaml_beginn= [email protected]" target=3D"_blank">[email protected]</a>&= gt;</span> wrote:<br><blockquote class=3D"gmail_quote" style=3D"border-left= :1px #ccc solid;"> <u></u> =20 <div style=3D"background-color:#fff;"> <span>=C2=A0</span> <div id=3D"m_-6420989893868275392ygrp-mlmsg"> <div id=3D"m_-6420989893868275392ygrp-msg"> <div id=3D"m_-6420989893868275392ygrp-text"> =20=20=20=20=20=20 =20=20=20=20=20=20 <p></p><div dir=3D"ltr"><font size=3D"4">Hi everyone,</font><div><fon= t size=3D"4"><br></font></div><div><font size=3D"4">This is a function I cr= eated to filter out all the prime numbers from 2 to some integer n.=C2=A0 I= like the function and it appears to be very fast and efficient.</font></di= v><div><font size=3D"4"><br></font></div><div><font size=3D"4"><div><b><fon= t color=3D"#4c1130">let sieve num =3D let rec sequence lower upper =3D if l= ower > upper then [ ] else lower :: sequence ( lower + 1 ) upper in=C2= =A0</font></b></div><div><b><font color=3D"#4c1130">=C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0let integers =3D sequence 2 num in=C2=A0</font></b></div><div><b><font c= olor=3D"#4c1130">=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0let num' =3D float_of_int num = in=C2=A0</font></b></div><div><b><font color=3D"#4c1130">=C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0let num'' =3D sqrt num' in=C2=A0</font></b></div><div><b>= <font color=3D"#4c1130">=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0let num''' =3D = ceil num'' in=C2=A0</font></b></div><div><b><font color=3D"#4c1130"= >=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0let num'''' =3D int_of_float num= 9;'' in=C2=A0</font></b></div><div><b><font color=3D"#4c1130">=C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0let rec sieve' start potential_primes =3D=C2=A0</fo= nt></b></div><div><b><font color=3D"#4c1130">=C2=A0 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0 if start > num'''' then potential_primes=C2= =A0</font></b></div><div><b><font color=3D"#4c1130">=C2=A0 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0 =C2=A0 else sieve' ( start + 1 ) ( List.filter ( fun x ->= if x > start && x mod start =3D 0 then false else true )=C2=A0<= /font></b></div><div><b><font color=3D"#4c1130">=C2=A0 =C2=A0 =C2=A0 =C2=A0= =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2= =A0 =C2=A0 potential_primes ) in=C2=A0</font></b></div><div><b><font color= =3D"#4c1130">=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0= =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 sieve' 2 integers ;;<= /font></b></div><div><b><font color=3D"#4c1130"><br></font></b></div><div><= b><font color=3D"#4c1130"><br></font></b></div><div><font color=3D"#000000"= >However, I'm having trouble with the reverse question.=C2=A0 I know fr= om experimentation that there are 25 prime integers in the interval [2, 100= ).=C2=A0 That's great.=C2=A0 I can find those integers by apply the fun= ction above with <b>sieve 100</b>.=C2=A0 But what a function called <b>find= NPrimes 25</b> that returns the first 25 prime numbers?=C2=A0 I'm findi= ng that function rather difficult to write.=C2=A0 Any clues or hints?=C2=A0= Thanks!!!</font></div><div><font color=3D"#000000"><br></font></div><div><= font color=3D"#000000">Best,</font></div><div><font color=3D"#000000"><br><= /font></div><div><font color=3D"#000000">Douglas.</font></div><div><font co= lor=3D"#000000"><br></font></div><div><font color=3D"#000000">P.S.=C2=A0 Do= es anyone know of any CS professors or professionals in the Chicago area wh= o are fairly adept at OCaml?=C2=A0 Also, I am curious about something.=C2= =A0 I read that OCaml really descended from a programming language called S= TANDARD ML.=C2=A0 How different is OCaml from Standard ML.=C2=A0 Pretty clo= se?=C2=A0 Or very different?=C2=A0 I appreciate the feedback.</font></div><= div><font color=3D"#000000"><br></font></div><div><font color=3D"#000000"><= br></font></div><div><font color=3D"#000000"><br></font></div><div><br></di= v></font></div></div> <p></p> </div> =20=20=20=20=20 =20=20=20=20 <div style=3D"color:#fff;height:0;"></div> </div> =20=20 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} #ygrp-mlmsg #logo { padding-bottom: 10px; } #ygrp-msg p a { font-family: Verdana; } #ygrp-msg p#attach-count span { color: #1E66AE; font-weight: 700; } #ygrp-reco #reco-head { color: #ff7900; font-weight: 700; } #ygrp-reco { margin-bottom: 20px; padding: 0px; } #ygrp-sponsor #ov li a { font-size: 130%; text-decoration: none; } #ygrp-sponsor #ov li { font-size: 77%; list-style-type: square; padding: 6px 0; }=20 #ygrp-sponsor #ov ul { margin: 0; padding: 0 0 0 8px; } #ygrp-text { font-family: Georgia; } #ygrp-text p { margin: 0 0 1em 0; } #ygrp-text tt { font-size: 120%; } #ygrp-vital ul li:last-child { border-right: none !important;=20 }=20 --> </style> </head> <!--~-|**|PrettyHtmlEnd|**|-~--> </html> <!-- end group email --> --001a1142dbde7378d60548a1deac--