A039672 -id:A039672 - cp's OEIS frontend

A000959 Lucky numbers.

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

Offset: 1

N. J. A. Sloane; entry updated Mar 07 2008

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 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-Keating (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
The Mersenne primes 2^p - 1 (= A000668, p in A000043) are in this sequence for p = 2, 3, 5, 7, 13, 17, and 19, but not for the following exponents p = 31, 61, and 89. - M. F. Hasler, May 06 2025

Martin Gardner, Gardner's Workout, Chapter 21 "Lucky Numbers and 2187" pp. 149-156 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 LA-3106, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Sep 1964.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 116.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 114.

Hugo van der Sanden, Table of n, a(n) for n = 1..200000 (terms 1..10000 from T. D. Noe, terms 10001..30981 from R. J. Mathar)
José Juan Barco, Highly efficient C implementation with O(n log n) time and O(n) space (n bits) using hierarchical Fenwick trees and bitmaps to optimize memory access. Based on: J. Bille, I. Larsen, I. L. Gørtz et al., "Succinct Partial Sums and Fenwick Trees," ESA 2017.
José Juan Barco, Very fast O(n log n) sequence generating program in Haskell
H. M. Bui and J. P. Keating, On twin primes associated with the Hawkins random sieve, J. Number Theory 119(2) (2006), 284-296.
H. M. Bui and J. P. Keating, On twin primes associated with the Hawkins random sieve, arXiv:math/0607196 [math.NT], 2006-2010.
Vema Gardiner, R. Lazarus, N. Metropolis, and S. Ulam, On certain sequences of integers defined by sieves, Math. Mag., 29 (1956), 117-122. doi:10.2307/3029719; Zbl 0071.27002.
Martin Gardner, Lucky numbers and 2187, Math. Intellig., 19(2) (1997), 26-29.
David 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, A Log Log Improvement to the Riemann Hypothesis for the Hawkins Random Sieve, Ann. Probability, 6 (1978), 850-875.
Ivars Peterson and 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. 6-7. This is Sieve #7. [Annotated and scanned copy]
Walter Schneider, Lucky Numbers, personal web site "Mathews: the Archive of Recreational Mathematics" (no more available: snapshot on web.archive.org as of May 2007), updated Dec 24, 2002.
Torsten Sillke, S. M. Ulam's Lucky Numbers
Hugo van der Sanden, Lucky numbers up to 1e8. [Broken link]
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 related to lucky numbers
Index entries for sequences generated by sieves [From _Reinhard Zumkeller_, Oct 15 2008]

Main diagonal of A258207.
Column 1 of A255545. (cf. also arrays A255543, A255551).
Cf. A050505 (complement).
Cf. A145649 (characteristic function).
Cf. A137164-A137185, A039672, A045954, A249876.
Cf. A031883 (first differences), A254967 (iterated absolute differences), see also A054978.
Cf. A109497 (works as a left inverse function).
The Gilbreath transform is A054978 - see also A362460, A362461, A362462.
Cf. also A000040, A003309, A032600, A219178, A255553, A264940, A265859.

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

-- Also see links.
(C++) // See Wilson link, Nov 14 2012

## 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 i, 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:
A000959List(10^3); # Robert Israel, Nov 19 2014

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 (* Jean-François Alcover, Mar 15 2016, after Robert Israel *)

A000959_upto(nMax)={my(v=vectorsmall(nMax\2,k,2*k-1),i=1,q);while(v[i++]<=#v,v=vecextract(v,2^#v-1-(q=1<M. F. Hasler, Sep 22 2013, improved Jan 20 2020

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
# Robert FERREOL, Nov 19 2014, corrected by F. Chapoton, Mar 29 2020, performance improved by Ely Golden, Aug 18 2022

(define (A000959 n) ((rowfun_n_for_A000959sieve n) n)) ;; Code for rowfun_n_for_A000959sieve given in A255543.
;; Antti Karttunen, Feb 26 2015

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) = A254967(n-1, n-1). - Reinhard Zumkeller, Feb 11 2015
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
A145649(a(n)) = 1; complement of A050505. - Reinhard Zumkeller, Oct 15 2008
Other identities 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)

A045954 Even-Lucky-Numbers: generated by a sieve process like that for Lucky numbers but starting with even numbers.

Original entry on oeis.org

2, 4, 6, 10, 12, 18, 20, 22, 26, 34, 36, 42, 44, 50, 52, 54, 58, 68, 70, 76, 84, 90, 98, 100, 102, 108, 114, 116, 118, 130, 132, 138, 140, 148, 150, 164, 170, 172, 178, 182, 186, 196, 198, 212, 214, 218, 228, 230, 234, 244, 246, 260, 262, 268, 278, 282, 290, 298, 300, 308

Offset: 1

Felice Russo

Write down even numbers: 2, 4, 6, 8, ...; first term > 2 is 4 so starting from 2 remove every 4th number: 2, 4, 6, 10, 12, 14, 18, ...; next number is 6 so remove every 6th term starting from 2: 2, 4, 6, 10, 12, 18, 20, 22, 26, etc.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for sequences generated by sieves

Cf. A000959, A039672.

a045954 n = a045954_list !! (n-1)
a045954_list =  2 : sieve 2 [2,4..] 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

## Finds all Even Lucky Numbers up to n from the list 2..n.
## Try n=10^5 or 10^6 just for fun!
evenluckynumbers:=proc(n) local k, Lnext, Lprev; Lprev:=[$2..n]; for k from 1 do Lnext:= map(w-> Lprev[w],remove(z -> z mod Lprev[k] = 0,[$1..nops(Lprev)])); if nops(Lnext)=nops(Lprev) then return Lnext fi; Lprev:=Lnext; od; end: # Walter Kehowski, Jun 06 2008

lst = Range[2, 308, 2]; i = 2; While[ i <= (len = Length@lst) && (k = lst[[i]]) <= len, lst = Drop[lst, {k, len, k}]; i++ ]; lst (* Robert G. Wilson v, May 11 2006 *)

A039673 Twin Fibonacci-lucky numbers.

Original entry on oeis.org

2, 4, 5, 7, 11, 13, 23, 25, 38, 40, 50, 52, 56, 58, 59, 61, 74, 76, 86, 88, 101, 103, 104, 106, 122, 124, 131, 133, 149, 151, 164, 166, 173, 175, 185, 187, 200, 202, 236, 238, 248, 250, 254, 256, 266, 268, 275, 277, 281, 283, 308, 310, 329, 331, 344, 346, 380

Offset: 1

Felice Russo

nonn

See A039672 for definition.

Cf. A039672 - Lower terms: A039693, upper terms: A039694, middle terms: A039695.

From R. J. Mathar, Oct 10 2010: (Start)
A039672 := proc(nmax) L := [seq(i,i=1..nmax)] ; for itr from 1 do if itr+1 > nops(L) then return L; end if; str := op(itr,L)+op(itr+1,L) ; for i from floor(nops(L)/str) to 1 by -1 do L := subsop(str*i=NULL,L) ; end do: print(L) ; end do: end proc:
L := A039672(600) ; for i from 1 to nops(L)-1 do if op(i,L) = op(i+1,L)-2 then printf("%d,%d,",op(i,L),op(i+1,L)) ; fi; end do: (End)

Corrected from a(25) on by R. J. Mathar, Oct 10 2010

A039694 Twin Fibonacci-lucky numbers (upper terms).

Original entry on oeis.org

4, 7, 13, 25, 40, 52, 58, 61, 76, 88, 103, 106, 124, 133, 151, 166, 175, 187, 202, 238, 250, 256, 268, 277, 283, 310, 331, 346, 382, 394, 427, 445, 463, 481, 493, 520, 529, 547, 571, 619, 634, 661, 772, 781, 796, 805, 817, 877, 886, 904, 949, 967, 985, 1006

Offset: 1

Felice Russo

nonn

See A039672 for definition.

A039673.

a(n) = A039693(n)+2. [From R. J. Mathar, Oct 10 2010]

Corrected by R. J. Mathar, Oct 10 2010

A039681 Odd Fibonacci-lucky numbers.

Original entry on oeis.org

1, 5, 7, 11, 13, 19, 23, 25, 29, 37, 41, 49, 59, 61, 65, 77, 83, 91, 97, 101, 103, 115, 121, 131, 133, 137, 149, 151, 155, 163, 173, 175, 181, 185, 187, 205, 211, 223, 227, 235, 247, 253, 275, 277, 281, 283, 295, 299, 319, 325, 329, 331, 347, 353, 367, 379, 389

Offset: 1

Felice Russo

nonn

See A039672 for definition.

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Feb 21 2001

A039695 Twin Fibonacci-lucky numbers (middle terms).

Original entry on oeis.org

3, 6, 12, 24, 39, 51, 57, 60, 75, 87, 102, 105, 123, 132, 150, 165, 174, 186, 201, 237, 249, 255, 267, 276, 282, 309, 330, 345, 381, 393, 426, 444, 462, 480, 492, 519, 528, 546, 570, 618, 633, 660, 771, 780, 795, 804, 816, 876, 885, 903, 948, 966, 984, 1005

Offset: 1

Felice Russo

nonn

See A039672 for definition.

A039673.

a(n) = A039693(n)+1 = A039694(n)-1. [From R. J. Mathar, Oct 10 2010]

Corrected by R. J. Mathar, Oct 10 2010

A039674 Palindromic Fibonacci-lucky numbers.

Original entry on oeis.org

1, 2, 4, 5, 7, 11, 77, 88, 101, 121, 131, 151, 181, 202, 212, 262, 353, 424, 464, 505, 515, 535, 545, 676, 707, 757, 787, 868, 919, 929, 949, 979, 1001, 1661, 2002, 2992, 4334, 5665, 5995, 6226, 8338, 8668, 8888, 9889, 10501, 10801, 11011, 11911, 12721, 13631

Offset: 1

Felice Russo

See A039672 for definition.

Corrected and extended by Naohiro Nomoto, Oct 28 2001

Single-digit palindromes and more terms from Sean A. Irvine, Feb 21 2021

A039675 Prime Fibonacci-lucky numbers.

Original entry on oeis.org

2, 5, 7, 11, 13, 19, 23, 29, 37, 41, 59, 61, 83, 97, 101, 103, 131, 137, 149, 151, 163, 173, 181, 211, 223, 227, 277, 281, 283, 331, 347, 353, 367, 379, 389, 397, 401, 421, 433, 443, 461, 463, 479, 491, 509, 541, 547, 563, 569, 571, 577, 601, 607, 613, 617

Offset: 1

Felice Russo

nonn

See A039672 for definition.

Corrected and extended by Naohiro Nomoto, Oct 28 2001

A039676 Differences between Fibonacci-lucky numbers.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 3, 3, 1, 3, 2, 3, 1, 3, 5, 1, 2, 1, 8, 1, 2, 4, 2, 1, 2, 4, 3, 6, 2, 1, 5, 1, 3, 2, 3, 6, 4, 2, 1, 2, 9, 3, 3, 1, 2, 6, 1, 2, 3, 1, 12, 2, 3, 1, 3, 5, 1, 2, 7, 2, 6, 1, 3, 2, 3, 6, 4, 2, 3, 3, 3, 1, 8, 3, 4, 8, 1, 2, 9, 1, 2, 3, 1, 2, 6, 4, 2, 7, 2, 3, 1, 2, 1

Offset: 1

Felice Russo

See A039672 for definition.

More terms from Naohiro Nomoto, Oct 28 2001

A039677 Partial sums of Fibonacci-lucky numbers.

Original entry on oeis.org

1, 3, 7, 12, 19, 29, 40, 53, 69, 88, 108, 131, 156, 184, 213, 245, 282, 320, 360, 401, 450, 500, 552, 608, 666, 725, 786, 851, 919, 993, 1069, 1146, 1228, 1311, 1397, 1485, 1576, 1673, 1774, 1877, 1981, 2087, 2202, 2320, 2441, 2563, 2687, 2817, 2948, 3081

Offset: 1

Felice Russo

nonn

See A039672 for definition.

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 23 2003

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A000959 Lucky numbers.

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A045954 Even-Lucky-Numbers: generated by a sieve process like that for Lucky numbers but starting with even numbers.

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A039673 Twin Fibonacci-lucky numbers.

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A039694 Twin Fibonacci-lucky numbers (upper terms).

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A039681 Odd Fibonacci-lucky numbers.

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A039695 Twin Fibonacci-lucky numbers (middle terms).

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A039674 Palindromic Fibonacci-lucky numbers.

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A039675 Prime Fibonacci-lucky numbers.

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A039676 Differences between Fibonacci-lucky numbers.

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A039677 Partial sums of Fibonacci-lucky numbers.

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