

A000959


Lucky numbers.
(Formerly M2616 N1035)


267



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
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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
Bui and Keating establish an asymptotic formula for the number of kdifference twin primes, and more generally to all ltuples, 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 BuiHeating (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
a(n) = A254967(n1,n1).  Reinhard Zumkeller, Feb 11 2015


REFERENCES

Martin Gardner, Gardner's Workout, Chapter 21 "Lucky Numbers and 2187" pp. 149156 A. K. Peters MA 2002.
Richard K. Guy, Unsolved Problems in Number Theory, C3.
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).
M. L. Stein and P. R. Stein, Tables of the Number of Binary Decompositions of All Even Numbers Less Than 200,000 into Prime Numbers and Lucky Numbers. Report LA3106, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Sep 1964.
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), 284296.
Vema Gardiner, R. Lazarus, N. Metropolis and S. Ulam, On certain sequences of integers defined by sieves, Math. Mag., 29 (1956), 117122. doi:10.2307/3029719. ISSN 0025570X. Zbl 0071.27002.
Martin Gardner, Lucky numbers and 2187, Math. Intellig., 19 (No. 2, 1997), 2629.
David Hawkins, The random sieve, Math. Mag. 31 (1958), 13.
D. Hawkins and W. E. Briggs, The lucky number theorem, Math. Mag. 31 1958 8184.
C. C. Heyde, A Log Log Improvement to the Riemann Hypothesis for the Hawkins Random Sieve, Ann. Probability, 6 (1978), 850875.
Ivars Peterson, MathTrek, Martin Gardner's Lucky Numbers (archived on Archive.org)
Ivars Peterson, Martin Gardner's Lucky Numbers (archived on Wikiwix.com)
Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 67. This is Sieve #7. [Annotated and scanned copy]
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 spaceefficient 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.
a(n) = A258207(n,n). [Where A258207 is a square array constructed from the numbers remaining after each step described above.]  Antti Karttunen, Aug 06 2015
Other identities:
A145649(a(n)) = 1; complement of A050505.  Reinhard Zumkeller, Oct 15 2008
From Antti Karttunen, Feb 26 2015: (Start)
For all n >= 1, A109497(a(n)) = n.
For all n >= 1, a(n) = A000040(n) + A032600(n).
For all n >= 2, a(n) = A255553(A000040(n)).
(End)


MAPLE

## luckynumbers(n) returns all lucky numbers from 1 to n. ## Try n=10^5 just for fun. luckynumbers:=proc(n) 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
# Alternative
L:= [seq(2*i+1, i=0..10^3)]:
for n from 2 while n < nops(L) do
r:= L[n];
L:= subsop(seq(r*i=NULL, i=1..nops(L)/r), L);
od:
L; # Robert Israel, Nov 19 2014


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
L = Table[2*i + 1, {i, 0, 10^3}]; For[n = 2, n < Length[L], r = L[[n++]]; L = ReplacePart[L, Table[r*i > Nothing, {i, 1, Length[L]/r}]]]; L (* JeanFrançois Alcover, Mar 15 2016, after Robert Israel *)


PROG

(Haskell)
a000959 n = a000959_list !! (n1)
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^#v1sum(j=1, #v\v[i], 2^(v[i]*j1)))); v} \\ M. F. Hasler, Sep 22 2013
(Python)
def lucky(n):
...L=list(range(1, n+1, 2)); j=1
...while L[j] <= len(L)1:
......L=[L[i] for i in range(len(L)) if (i+1)%L[j]!=0]
......j+=1
...return(L)
Robert FERREOL, Nov 19 2014
(Scheme)
(define (A000959 n) ((rowfun_n_for_A000959sieve n) n)) ;; Code for rowfun_n_for_A000959sieve given in A255543.
;; Antti Karttunen, Feb 26 2015


CROSSREFS

Main diagonal of A258207.
Column 1 of A255545. (cf. also arrays A255543, A255551).
Cf. A050505 (complement).
Cf. A145649 (characteristic function).
Cf. A137164A137185, A039672, A045954, A249876.
Cf. A031883 (first differences), A254967 (iterated absolute differences), see also A054978.
Cf. A109497 (works as an left inverse function).
Cf. also A000040, A003309, A032600, A219178, A255553, A264940, A265859.
Sequence in context: A172367 A024901 A258011 * 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


EXTENSIONS

Typo in Walter Kehowski's Maple program fixed by Robert Israel, Nov 19 2014


STATUS

approved



