A003309 Ludic numbers: apply the same sieve as Eratosthenes, but cross off every k-th remaining number. (Formerly M0655)

1, 2, 3, 5, 7, 11, 13, 17, 23, 25, 29, 37, 41, 43, 47, 53, 61, 67, 71, 77, 83, 89, 91, 97, 107, 115, 119, 121, 127, 131, 143, 149, 157, 161, 173, 175, 179, 181, 193, 209, 211, 221, 223, 227, 233, 235, 239, 247, 257, 265, 277, 283, 287, 301, 307, 313

Offset

1,2

Comments

The definition can obviously only be applied from k = a(2) = 2 on: for k = 1, all remaining numbers would be deleted. - M. F. Hasler, Nov 02 2024

References

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

Links

Donovan Johnson, Table of n, a(n) for n = 1..100000

David Applegate, C program for A003309.

Donovan Johnson, Ludic numbers computed up to A003309(1236290) = 23000711.

OEIS Wiki, Ludic numbers.

Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. This is Sieve #1. [Annotated and scanned copy]

Rosettacode Wiki, Ludic numbers.

Index entries for sequences generated by sieves

Formula

Complement of A192607; A192490(a(n)) = 1. - Reinhard Zumkeller, Jul 05 2011

From Antti Karttunen, Feb 23 2015: (Start)

a(n) = A255407(A008578(n)).

a(n) = A008578(n) + A255324(n).

(End)

Maple

ludic:= proc(N) local i, k, S, R;
  S:= {$2..N};
  R:= 1;
  while nops(S) > 0 do
    k:= S[1];
    R:= R, k;
    S:= subsop(seq(1+k*j=NULL, j=0..floor((nops(S)-1)/k)), S);
  od:
[R];
end proc:
ludic(1000); # Robert Israel, Feb 23 2015

Mathematica

t = Range[2, 400]; r = {1}; While[Length[t] > 0, k = First[t]; AppendTo[r, k]; t = Drop[t, {1, -1, k}]; ]; r (* Ray Chandler, Dec 02 2004 *)

Prog

(PARI) t=vector(399, x, x+1); r=[1]; while(length(t)>0, k=t[1]; r=concat(r, [k]); t=vector((length(t)*(k-1))\k, x, t[(x*k+k-2)\(k-1)])); r \\ Phil Carmody, Feb 07 2007
(PARI) A3309=[1]; next_A003309(n)=n<A3309[#A3309] || until(, my(k=(#A3309-1)\2); forstep(j=#A3309-k, 2, -1, k+=1+k\(A3309[j]-1)); A3309=concat(A3309, k+2); k+2>n && break); n+!if(n=setsearch(A3309, n+1, 1), return(A3309[n])) \\ Should be made more efficient if n >> max(A3309). - M. F. Hasler, Nov 02 2024
{A003309(n) = while(n>#A3309, next_A003309(A3309[#A3309])); A3309[n]} \\ Should be made more efficient in case n >> #A3309. - M. F. Hasler, Nov 03 2024
(Haskell)
a003309 n = a003309_list !! (n - 1)
a003309_list = 1 : f [2..] :: [Int]
   where f (x:xs) = x : f (map snd [(u, v) | (u, v) <- zip [1..] xs,
                                             mod u x > 0])
-- Reinhard Zumkeller, Feb 10 2014, Jul 03 2011
(Scheme)
(define (A003309 n) (if (= 1 n) n (A255127bi (- n 1) 1))) ;; Code for A255127bi given in A255127.
;; Antti Karttunen, Feb 23 2015
(Python)
remainders = [0]
ludics = [2]
N_MAX = 313
for i in range(3, N_MAX) :
    ludic_index = 0
    while ludic_index < len(ludics) :
        ludic = ludics[ludic_index]
        remainder = remainders[ludic_index]
        remainders[ludic_index] = (remainder + 1) % ludic
        if remainders[ludic_index] == 0 :
            break
        ludic_index += 1
    if ludic_index == len(ludics) :
        remainders.append(0)
        ludics.append(i)
ludics = [1] + ludics
print(ludics)
# Alexandre Herrera, Aug 10 2023
def A003309(): # generator of the infinite list of ludic numbers
    L = [2, 3]; yield 1; yield 2; yield 3
    while k := len(L)//2: # could take min{k | k >= L[-1-k]-1}
        for j in L[-1-k::-1]: k += 1 + k//(j-1)
        L.append(k+2); yield k+2
A003309_upto = lambda N=99: [t for t, _ in zip(A003309(), range(N))]
# M. F. Hasler, Nov 02 2024

Crossrefs

Without the initial 1 occurs as the leftmost column in arrays A255127 and A260717.

Cf. A003310, A003311, A100464, A100585, A100586 (variants).

Cf. A192503 (primes in sequence), A192504 (nonprimes), A192512 (number of terms <= n).

Cf. A192490 (characteristic function).

Cf. A192607 (complement).

Cf. A260723 (first differences).

Cf. A255420 (iterates of f(n) = A003309(n+1) starting from n=1).

Subsequence of A302036.

Cf. A237056, A237126, A237427, A235491, A255407, A255408, A255421, A255422, A260435, A260436, A260741, A260742 (permutations constructed from Ludic numbers).

Cf. also A000959, A008578, A255324, A254100, A272565 (Ludic factor of n), A297158, A302032, A302038.

Cf. A376237 (ludic factorial: cumulative product), A376236 (ludic Fortunate numbers).

Sequence in context: A198196 A139054 A290959 * A063884 A316787 A165671

Adjacent sequences: A003306 A003307 A003308 * A003310 A003311 A003312

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from David Applegate and N. J. A. Sloane, Nov 23 2004

Status

approved