login
This site is supported by donations to The OEIS Foundation.

 

Logo

110 people attended OEIS-50 (videos, suggestions); annual fundraising drive to start soon (donate); editors, please edit! (stack is over 300), your editing is more valuable than any donation.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000959 Lucky numbers.
(Formerly M2616 N1035)
221
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, 105, 111, 115, 127, 129, 133, 135, 141, 151, 159, 163, 169, 171, 189, 193, 195, 201, 205, 211, 219, 223, 231, 235, 237, 241, 259, 261, 267, 273, 283, 285, 289, 297, 303 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

A145649(a(n)) = 1; complement of A050505. - Reinhard Zumkeller, Oct 15 2008

Bui and Keating establish an asymptotic formula for the number of k-difference twin primes, and more generally to all l-tuples, of Hawkins primes, a probabilistic model of the Eratosthenes sieve. The formula for k = 1 was obtained by Wunderlich [Acta Arith. 26 (1974), 59 - 81]. - Jonathan Vos Post, Mar 24 2009. (This is quoted from the abstract of the Bui-Heating (2006) article, Joerg Arndt, Jan 04 2014)

It appears that a 1's line is formed, as in the Gilbreath's conjecture, if we use 2 (or 4), 3, 5 (differ of 7), 9, 13, 15, 21, 25,... instead of A000959 1, 3, 7, 9, 13, 15, 21, 25, ... - Eric Desbiaux, Mar 25 2010

REFERENCES

Martin Gardner, Lucky numbers and 2187, Math. Intellig., 19 (No. 2, 1997), 26-29.

Martin Gardner, Gardner's Workout, Chapter 21 "Lucky Numbers and 2187" pp. 149-156 A. K. Peters MA 2002.

Vema Gardiner, R. Lazarus, N. Metropolis and Stanislaw Ulam, On certain sequences of integers defined by sieves, Math. Mag., 29 (1955), 117-119.

Richard K. Guy, Unsolved Problems in Number Theory, C3.

D. Hawkins, The random sieve, Math. Mag. 31 (1958), 1-3.

D. Hawkins and W. E. Briggs, The lucky number theorem. Math. Mag. 31 1958 81-84.

C. C. Heyde, Ann. Probability, 6 (1978), 850-875.

C. S. Ogilvy, Tomorrow's Math. 2nd ed., Oxford Univ. Press, 1972, p. 99.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 114.

LINKS

T. D. Noe, R. J. Mathar, and Hugo v. d. Sanden, Table of n, a(n) for n = 1..200000 (first 10000 terms from T. D. Noe. Terms up to 30981 by R. J. Mathar)

H. M. Bui, J. P. Keating, On twin primes associated with the Hawkins random sieve, version 2, Mar 24, 2009. J. Number Theory 119 (2006), 284-296.

Ivars Peterson, MathTrek, Martin Gardner's Lucky Numbers (archived on Archive.org)

Ivars Peterson, Martin Gardner's Lucky Numbers (archived on Wikiwix.com)

Walter Schneider, Lucky Numbers

Torsten Sillke, S. M. Ulam's Lucky Numbers

Hugo van der Sanden, Lucky numbers up to 1e8

G. Villemin's Almanach of Numbers, Nombre Chanceux

Eric Weisstein's World of Mathematics, Lucky number.

Wikipedia, Lucky number

David W. Wilson, Fast space-efficient sequence generating program in C++

Index entries for "core" sequences

Index entries for sequences generated by sieves [From Reinhard Zumkeller, Oct 15 2008]

FORMULA

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.

MAPLE

## 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 Kehowski, Jun 05 2008

MATHEMATICA

luckies = 2*Range@200 - 1; f[n_] := Block[{k = luckies[[n]]}, luckies = Delete[luckies, Table[{k}, {k, k, Length@luckies, k}]]]; Do[f@n, {n, 2, 30}]; luckies (* Robert G. Wilson v, May 09 2006 *)

sieveMax = 10^6; luckies = Range[1, sieveMax, 2]; sieve[n_] := Module[{k = luckies[[n]]}, luckies = Delete[luckies, Table[{i}, {i, k, Length[luckies], k}]]]; n = 1; While[luckies[[n]] < Length[luckies], n++; sieve[n]]; luckies

PROG

(Haskell)

a000959 n = a000959_list !! (n-1)

a000959_list =  1 : sieve 2 [1, 3..] where

   sieve k xs = z : sieve (k + 1) (lucky xs) where

      z = xs !! (k - 1 )

      lucky ws = us ++ lucky vs where

            (us, _:vs) = splitAt (z - 1) ws

-- Reinhard Zumkeller, Dec 05 2011

(PARI) A000959(nMax)={my(v=vector(nMax, i, i), i, k); while(v[i=!k+k++]<=#v, v=vecextract(v, 2^#v-1-sum(j=1, #v\v[i], 2^(v[i]*j-1)))); v} \\ - M. F. Hasler, Sep 22 2013

CROSSREFS

Cf. A137164-A137185, A039672, A045954.

Sequence in context: A073671 A172367 A024901 * A204085 A230076 A120226

Adjacent sequences:  A000956 A000957 A000958 * A000960 A000961 A000962

KEYWORD

nonn,easy,nice,core

AUTHOR

N. J. A. Sloane. Entry updated Mar 07 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified October 26 02:56 EDT 2014. Contains 248566 sequences.