cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A000959 Lucky numbers.

Original entry on oeis.org

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

Views

Author

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

Keywords

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 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

References

  • 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.

Links

Crossrefs

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.

Programs

  • 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
  • Haskell
    -- Also see links.
(C++) // See Wilson link, Nov 14 2012
  • 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 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
  • 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 (* Jean-François Alcover, Mar 15 2016, after Robert Israel *)
  • PARI
    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
  • 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
# Robert FERREOL, Nov 19 2014, corrected by F. Chapoton, Mar 29 2020, performance improved by Ely Golden, Aug 18 2022
  • Scheme
    (define (A000959 n) ((rowfun_n_for_A000959sieve n) n)) ;; Code for rowfun_n_for_A000959sieve given in A255543.
;; Antti Karttunen, Feb 26 2015

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) = 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)

A272565 Smallest ludic factor of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 5, 2, 3, 2, 23, 2, 25, 2, 3, 2, 29, 2, 7, 2, 3, 2, 5, 2, 37, 2, 3, 2, 41, 2, 43, 2, 3, 2, 47, 2, 5, 2, 3, 2, 53, 2, 11, 2, 3, 2, 7, 2, 61, 2, 3, 2, 5, 2, 67, 2, 3, 2, 71, 2, 13, 2, 3, 2, 77, 2, 5, 2, 3
Offset: 1

Views

Author

Max Barrentine, May 09 2016

Keywords

Comments

This sequence is somewhat analogous to the smallest prime factor of n (A020639). However, each natural number has only one ludic factor, because once it is crossed off in the k-th step of the sieve process, it is not a member of the terms considered in the (k+1)-th step.
On the other hand, by iteratively invoking A302032 it is possible to factor n to its constituent "Ludic factors", with each natural number having a unique such decomposition, analogous to prime factorization of n. See comments and examples given in A302032. - Antti Karttunen, Apr 08 2018
The "ludic factor" here is the k which either yields one of the ludic numbers A003309, or is used to cross out a non ludic number. In that case, this "ludic factor" often does not divide n, see A276569. But in the usual sieve of Eratosthenes, the fact that numbers are crossed out from the list does not mean they don't have other factors, so exactly the same could be considered here, which makes disputable the assertion that numbers have only one ludic factor. - M. F. Hasler, Nov 03 2024

Links

Crossrefs

Cf. A003309 (ludic numbers), A020639 (least prime factor), A027748 (prime factors of n), A192607, A255127, A260738, A276440, A276568, A276569, A302032.
Cf. A276347, A276447, A276448 (ludic factor is equal, less than or greater than the smallest prime factor).
Cf. A260739 (ordinal transform), A302036 (numbers with all Ludic factors equal).
Cf. A264940 (analogous version for lucky numbers).

Programs

Formula

From Antti Karttunen, Sep 11 2016: (Start)
a(n) = A003309(1 + A260738(n)).
For all n >= 1, a(A276347(n)) = A020639(A276347(n)). (End)
From M. F. Hasler, Nov 04 2024: (Start)
To rephrase the above: By definition, k is in A276347 iff a(k) = A020639(k).
Particular cases: a(2n) = 2 and a(6n-3) = 3 for all n. (End)

Extensions

Added "smallest" in the definition because the explanation of "only one..." in the first comment might be disputable. - M. F. Hasler, Nov 03 2024

A271419 If n is a ludic number, a(n)=0; if n is not a ludic number, a(n) is the ludic number that rejects n from the ludic number sieve.

Original entry on oeis.org

0, 0, 0, 2, 0, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 5, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 7, 2, 3, 2, 5, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 5, 2, 3, 2, 0, 2, 11, 2, 3, 2, 7, 2, 0, 2, 3, 2, 5, 2, 0, 2, 3, 2, 0, 2, 13, 2, 3, 2, 0, 2, 5, 2, 3, 2, 0, 2, 7
Offset: 1

Views

Author

Max Barrentine, Apr 07 2016

Keywords

Links

Crossrefs

Cf. A003309, A192607, A192490, A255127, A260738, A271420, A272565.
Cf. A264940 (analogous version for lucky numbers).

Programs

Formula

a(n) = (1-A192490(n)) * A272565(n). - Antti Karttunen, Sep 11 2016

A274568 Irregular triangle read by rows in which the n-th row (n>1) lists distinct "factors" of n for the sieve described in A262775.

Original entry on oeis.org

2, 0, 2, 3, 7, 2, 3, 2, 9, 2, 3, 13, 2, 7, 3, 2, 15, 2, 3, 19, 2, 9, 3, 7, 2, 21, 2, 3, 27, 2, 13, 3, 2, 7, 9, 2, 3, 33, 2, 15, 3, 2, 7, 2, 3, 37, 2, 9, 19, 3, 13, 2, 39, 2, 3, 7, 49, 2, 21, 3, 2, 9, 15, 2, 3, 7, 2, 3, 2, 13, 27, 51, 2, 3, 55, 2, 7, 9, 3, 19
Offset: 0

Views

Author

Max Barrentine, Jun 28 2016

Keywords

Comments

This sequence is analogous to the prime factors of n, but for the sifted set A262775. If n is equivalent to some term A262775(k) mod A262775(k+1), then row n will include the term A262775(k+1). Row 1 is zero.

Examples

			Row 2 is 2,3 because 2==0 mod 2 and A262775(0)=0, A262775(1)=2, and 2==2 mod 3 and A262775(1)=2, A262775(2)=3.

Links

Crossrefs

Cf. A262775, A027748, A264940, A272565.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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