

A000959


Lucky numbers.
(Formerly M2616 N1035)


304



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
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


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

Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 67. This is Sieve #7. [Annotated and scanned copy]


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
(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; typo fixed by Robert Israel, Nov 19 2014
# Alternative
A000959List := proc(mx) local L, n, r;
L:= [seq(2*i+1, i=0..mx)]:
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 end:


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
(C++) // See Wilson link, Nov 14 2012
(PARI) A000959_upto(nMax)={my(v=vectorsmall(nMax\2, k, 2*k1), i=1, q); while(v[i++]<=#v, v=vecextract(v, 2^#v1(q=1<<v[i])^(#v\v[i])\(q1)<<(v[i]1) )); v} \\ M. F. Hasler, Sep 22 2013, improved Jan 20 2020
(Python)
def lucky(n):
L = list(range(1, n + 1, 2))
j = 1
while j <= len(L)  1 and L[j] <= len(L):
del L[L[j]1::L[j]]
j += 1
return L
(Scheme)
(define (A000959 n) ((rowfun_n_for_A000959sieve n) n)) ;; Code for rowfun_n_for_A000959sieve given in A255543.


CROSSREFS

Cf. A145649 (characteristic function).
Cf. A109497 (works as a left inverse function).


KEYWORD

nonn,easy,nice,core


AUTHOR



STATUS

approved



