%I A000959 M2616 N1035
%S A000959 1,3,7,9,13,15,21,25,31,33,37,43,49,51,63,67,69,73,75,79,87,93,99,
%T A000959 105,111,115,127,129,133,135,141,151,159,163,169,171,189,193,195,201,
%U A000959 205,211,219,223,231,235,237,241,259,261,267,273,283,285,289,297,303
%N A000959 Lucky numbers.
%C A000959 An interesting general discussion of the phenomenon of 'random primes' 
               (generalizing the lucky numbers) occurs in Hawkins (1958). Heyde 
               (1978) proves that Hawkins' random primes do not only almost always 
               satisfy the Prime Number Theorem but also the Riemann Hypothesis. 
               - Alf van der Poorten, Jun 27 2002
%C A000959 A145649(a(n)) = 1; complement of A050505. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Oct 15 2008]
%C A000959 Bui and Keating establish an asymptotic formula for the number of k-difference 
               twin primes associated with the Hawkins random sieve, which is a 
               probabilistic model of the Eratosthenes sieve. The formula for k 
               = 1 was obtained by Wunderlich [Acta Arith. 26 (1974), 59 - 81]. 
               We here extend this to k => 2 and generalize it to all l-tuples of 
               Hawkins primes. [From Jonathan Vos Post (jvospost3(AT)gmail.com), 
               Mar 24 2009]
%D A000959 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000959 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000959 M. Gardner, Lucky numbers and 2187, Math. Intellig., 19 (No. 2, 1997), 
               26-29.
%D A000959 M. Gardner, Gardner's Workout, Chapter 21 "Lucky Numbers and 2187" pp. 
               149-156 A. K. Peters MA 2002.
%D A000959 V. Gardiner, R. Lazarus, N. Metropolis and S. Ulam, On certain sequences 
               of integers defined by sieves, Math. Mag., 29 (1955), 117-119.
%D A000959 R. K. Guy, Unsolved Problems in Number Theory, C3.
%D A000959 D. Hawkins, The random sieve, Math. Mag. 31 (1958), 1-3.
%D A000959 D. Hawkins and W. E. Briggs, The lucky number theorem. Math. Mag. 31 
               1958 81-84.
%D A000959 C. C. Heyde, Ann. Probability, 6 (1978), 850-875.
%D A000959 C. S. Ogilvy, Tomorrow's Math. 2nd ed., Oxford Univ. Press, 1972, p. 
               99.
%D A000959 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. 
               Penguin Books, NY, 1986, 114.
%H A000959 R. J. Mathar, <a href="b000959.txt">Table of n, a(n) for n = 1..30981</
               a>
%H A000959 I. Peterson, MathTrek, <a href="http://www.maa.org/mathland/mathtrek1.html">
               Martin Gardner's Lucky Numbers</a>
%H A000959 I. Peterson, <a href="http://www.sciencenews.org/sn_arc97/9_6_97/mathland.htm">
               See also</a>
%H A000959 W. Schneider, <a href="http://www.wschnei.de/number-theory/lucky-numbers.html">
               Lucky Numbers</a> [Broken link?]
%H A000959 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/
               series013">S.M.Ulam's Lucky Numbers</a>
%H A000959 G. Villemin's Almanach of Numbers, <a href="http://www.multimania.com/
               villemingerard/Iteration/Chanceux.htm">Nombre Chanceux</a>
%H A000959 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LuckyNumber.html">Lucky number.</a>
%H A000959 Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucky_number">Lucky 
               number</a>
%H A000959 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000959 <a href="Sindx_Si.html#sieve">Index entries for sequences generated by 
               sieves</a> [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Oct 15 2008]
%H A000959 H. M. Bui, J. P. Keating, <a href="http://arxiv.org/abs/math/0607196">
               On twin primes associated with the Hawkins random sieve</a>, version 
               2, Mar 24, 2009. J. Number Theory 119 (2006), 284-296. [From Jonathan 
               Vos Post (jvospost3(AT)gmail.com), Mar 24 2009]
%F A000959 Start with the natural numbers. Delete every 2nd number, leaving 1 3 
               5 7 ...; the 2nd number remaining is 3, so delete every 3rd number, 
               leaving 1 3 7 9 13 15 ...; now delete every 7th number, leaving 1 
               3 7 9 13 ...; now delete every 9th number; etc.
%p A000959 ## luckynumbers(n) returns all lucky numbers from 1 to n. ## Try n=10^5 
               just for fun. luckynumbers:=proc(L) local k, Lnext, Lprev; Lprev:=[$1..n]; 
               for k from 1 do if k=1 or k=2 then Lnext:= map(w-> Lprev[w],remove(z 
               -> z mod Lprev[2] = 0,[$1..nops(Lprev)])); if nops(Lnext)=nops(Lprev) 
               then break fi; Lprev:=Lnext; else Lnext:= map(w-> Lprev[w],remove(z 
               -> z mod Lprev[k] = 0,[$1..nops(Lprev)])); if nops(Lnext)=nops(Lprev) 
               then break fi; Lprev:=Lnext; fi; od; return Lnext; end: - Walter 
               A. Kehowski (wkehowski(AT)cox.net), Jun 05 2008
%t A000959 t = 2Range@200 - 1; f[n_] := Block[{k = t[[n]]}, t = Delete[t, Table[{k}, 
               {k, k, Length@t, k}]]]; Do[f@n, {n, 2, 30}]; t (from Robert G. Wilson 
               v (rgwv(at)rgwv.com), May 09 2006)
%Y A000959 Cf. A137164-A137185.
%Y A000959 Sequence in context: A032678 A073671 A024901 this_sequence A120226 A137310 
               A118567
%Y A000959 Adjacent sequences: A000956 A000957 A000958 this_sequence A000960 A000961 
               A000962
%K A000959 nonn,easy,nice,core
%O A000959 1,2
%A A000959 N. J. A. Sloane (njas(AT)research.att.com). Entry updated Mar 07 2008