A100464 -id:A100464 - cp's OEIS frontend

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

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

N. J. A. Sloane

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

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

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

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).

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

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

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 *)

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

A3309=[1]; next_A003309(n)=nn && 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

upto(nn)= my(r=List([1..nn]), p=1); while(p++<#r, my(k=r[p], i=p); while((i+=k)<=#r, listpop(~r, i); i--)); Vec(r); \\ Ruud H.G. van Tol, Dec 13 2024

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

(define (A003309 n) (if (= 1 n) n (A255127bi (- n 1) 1))) ;; Code for A255127bi given in A255127.
;; Antti Karttunen, Feb 23 2015

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)

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

A100585 a(n+1) = a(n)+floor(a(n)/3), a(1) = 3.

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 13, 17, 22, 29, 38, 50, 66, 88, 117, 156, 208, 277, 369, 492, 656, 874, 1165, 1553, 2070, 2760, 3680, 4906, 6541, 8721, 11628, 15504, 20672, 27562, 36749, 48998, 65330, 87106, 116141, 154854, 206472, 275296, 367061, 489414, 652552

Offset: 1

N. J. A. Sloane, Dec 01 2004

nonn

Original definition: Write down the numbers from 3 to infinity. Take next number, M say, that has not been crossed off. Counting through the numbers that have not yet been crossed off after that M, cross off every 4th term. Repeat, always crossing off every 4th term of those that remain. The numbers that are left form the sequence.

Can be stated as the number of animals starting from a single trio if any trio of animals can produce a single offspring. See A061418 for the equivalent sequence for pairs of animals. - Luca Khan, Sep 05 2024

Robert Israel, Table of n, a(n) for n = 1..7987
Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. This is Sieve #6 with K=4. [Annotated and scanned copy]
Index entries for sequences generated by sieves

Cf. A003309, A003310, A100464, A100562, A003312, A003311, A052548, A100586, A061418.

R:= 3: x:= 3:
for i from 2 to 100 do x:= x + floor(x/3); R:= R,x od:
R; # Robert Israel, Sep 09 2024

t = Range[3, 2500000]; r = {}; While[Length[t] > 0, AppendTo[r, First[t]]; t = Drop[t, {1, -1, 4}];]; r (* Ray Chandler, Dec 02 2004 *)
NestList[#+Floor[#/3]&,3,50] (* Harvey P. Dale, Jan 14 2019 *)

a(n,s=3)=for(i=2,n,s+=s\3);s \\ M. F. Hasler, Oct 06 2014

a(1)=3, a(n+1) = a(n) + floor(a(n)/3). - Ben Paul Thurston, Jan 09 2008

More terms from Ray Chandler, Dec 02 2004

Simpler definition from M. F. Hasler, Oct 06 2014

A003310 Generated by a sieve.

Original entry on oeis.org

3, 4, 5, 7, 8, 11, 13, 17, 19, 20, 26, 29, 32, 37, 38, 43, 49, 50, 56, 62, 67, 68, 71, 73, 86, 89, 91, 98, 103, 113, 116, 121, 127, 131, 133, 137, 140, 151, 158, 161, 169, 173, 179, 182, 188, 200, 206, 209, 211, 221, 227, 230, 239, 242, 247, 253, 259, 271, 277, 278

Offset: 1

N. J. A. Sloane

Apply the sieve of A003309, but begin with 3 rather than 2.

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. This is Sieve #2. [Annotated and scanned copy]
Index entries for sequences generated by sieves

Cf. A003309, A100464.

a003310 n = a003310_list !! (n-1)
a003310_list = f [3..] where
   f (x:xs) = x : f (g xs) where
     g zs = us ++ g vs where (us, _:vs) = splitAt (x - 1) zs
-- Reinhard Zumkeller, Nov 12 2014

t = Range[3, 330]; r = {}; While[Length[t] >0, k = First[t]; AppendTo[r, k]; t = Drop[t, {1, -1, k}];]r (* Ray Chandler, Dec 02 2004 *)

More terms from Ray Chandler, Dec 02 2004

A003312 a(1) = 3; for n>0, a(n+1) = a(n) + floor((a(n)-1)/2).

Original entry on oeis.org

3, 4, 5, 7, 10, 14, 20, 29, 43, 64, 95, 142, 212, 317, 475, 712, 1067, 1600, 2399, 3598, 5396, 8093, 12139, 18208, 27311, 40966, 61448, 92171, 138256, 207383, 311074, 466610, 699914, 1049870, 1574804, 2362205, 3543307, 5314960, 7972439, 11958658, 17937986, 26906978

Offset: 1

N. J. A. Sloane

This sequence was originally defined in Popular Computing in 1974 by a sieve, as follows. Write down the numbers from 3 to infinity. Take next number, M say, that has not been crossed off. Counting through the numbers that have not yet been crossed off after that M, cross off every third term. Repeat, always crossing off every third term of those that remain. The numbers that are left form the sequence. The recurrence given here for the sequence was found by Colin Mallows. The problem asked for the 1000th term, and was unsolved for several years.

			The first few sieving stages are as follows:
  3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...
  3 4 5 X 7 8 X 10 11 XX 13 14 XX 16 17 XX 19 20 ...
  3 4 5 X 7 X X 10 11 XX XX 14 XX 16 XX XX 19 20 ...
  3 4 5 X 7 X X 10 XX XX XX 14 XX 16 XX XX XX 20 ...
  3 4 5 X 7 X X 10 XX XX XX 14 XX XX XX XX XX 20 ...

Popular Computing (Calabasas, CA), Problem 43, Sieves, sieve #5, Vol. 2 (No. 13, Apr 1974), pp. 6-7; Vol. 2 (No. 17, Aug 1974), page 16; Vol. 5 (No. 51, Jun 1977), p. 17.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..1000 (first 500 terms from T. D. Noe)
Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. This is Sieve #5. [Annotated and scanned copy]
H. P. Robinson, C. L. Mallows, & N. J. A. SloaneCorrespondence, 1975
Index entries for sequences generated by sieves

Cf. A003309, A003310, A100464, A100562, A006999, A061418, A070885, A003311.

a003312 n = a003312_list !! (n-1)
a003312_list = sieve [3..] where
   sieve :: [Integer] -> [Integer]
   sieve (x:xs) = x : (sieve $ xOff xs)
   xOff :: [Integer] -> [Integer]
   xOff (x:x':_:xs) = x : x': (xOff xs)
-- Reinhard Zumkeller, Feb 21 2011

f:=proc(n) option remember; if n=1 then RETURN(3) fi; f(n-1)+floor( (f(n-1)-1)/2 ); end;

NestList[#+Floor[(#-1)/2]&,3,50]  (* Harvey P. Dale, Mar 18 2011 *)

v=vector(100); v[1]=3; for(n=2, #v, v[n]=floor((3*v[n-1]-1)/2)); v \\ Clark Kimberling, Dec 30 2010

l=[0, 3]
for n in range(2, 101):
    l.append(l[n - 1] + (l[n - 1] - 1)//2)
print(l[1:]) # Indranil Ghosh, Jun 09 2017

from itertools import islice
def A003312_gen(): # generator of terms
    a = 3
    while True:
        yield a
        a += a-1>>1
A003312_list = list(islice(A003312_gen(),30)) # Chai Wah Wu, Sep 21 2022

Entry revised by N. J. A. Sloane, Dec 01 2004 and May 10 2015

A003311 Write down the numbers from 3 to infinity. Take next number, M say, that has not been crossed off. Counting through the numbers that have not yet been crossed off after that M, cross off the first, (M+1)st, (2M+1)st, (3M+1)st, etc. Repeat. The numbers that are left form the sequence.

Original entry on oeis.org

3, 5, 8, 11, 15, 18, 23, 27, 32, 38, 42, 47, 53, 57, 63, 71, 75, 78, 90, 93, 98, 105, 113, 117, 123, 132, 137, 140, 147, 161, 165, 168, 176, 183, 188, 197, 206, 212, 215, 227, 233, 237, 243, 252, 258, 267, 278, 282, 287, 293, 303, 312, 317, 323

Offset: 1

N. J. A. Sloane

nonn

			The first few sieving stages are as follows:
  3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...
  3 X 5 6 X 8 9 XX 11 12 XX 14 15 XX 17 18 XX 20 ...
  3 X 5 X X 8 9 XX 11 12 XX XX 15 XX 17 18 XX 20 ...
  3 X 5 X X 8 X XX 11 12 XX XX 15 XX 17 18 XX 20 ...
  3 X 5 X X 8 X XX 11 XX XX XX 15 XX 17 18 XX 20 ...
  3 X 5 X X 8 X XX 11 XX XX XX 15 XX XX 18 XX 20 ...

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. Based on a misreading of Sieve #3. A100464 is the correct version. [Annotated and scanned copy]
Index entries for sequences generated by sieves

Cf. A003309, A003310, A100464, A003312, A100562, A100585, A052548.

a003311 n = a003311_list !! (n-1)
a003311_list = f [3..] where
   f (x:xs) = x : f (g xs) where
     g zs = us ++ g vs where (_:us, vs) = splitAt x zs
-- Reinhard Zumkeller, Nov 12 2014

Entry revised Nov 29 2004

A100562 Write down the numbers from 7 to infinity. Take next number, M say, that has not been crossed off and cross off all the numbers i*M - 1 for i >= 2. Repeat. The numbers that are left form the sequence.

Original entry on oeis.org

7, 8, 9, 10, 11, 12, 14, 16, 18, 22, 24, 25, 28, 30, 33, 36, 37, 38, 40, 42, 45, 46, 50, 51, 52, 56, 57, 58, 60, 61, 64, 66, 67, 68, 70, 72, 77, 78, 81, 82, 84, 85, 86, 88, 92, 93, 94, 96, 100, 102, 105, 106, 108, 112, 114, 117, 122, 123, 126, 128, 130, 136

Offset: 1

N. J. A. Sloane, Nov 23 2004

nonn

Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. This is Sieve #4. [Annotated and scanned copy]
Index entries for sequences generated by sieves

Cf. A003309, A003310, A100464.

A100586 Write down the numbers from 3 to infinity. Take next number, M say, that has not been crossed off. Counting through the numbers that have not yet been crossed off after that M, cross off every 5th term. Repeat, always crossing off every 5th term of those that remain. The numbers that are left form the sequence.

Original entry on oeis.org

3, 4, 5, 6, 7, 9, 11, 14, 17, 21, 26, 32, 40, 50, 62, 77, 96, 120, 150, 187, 234, 292, 365, 456, 570, 712, 890, 1112, 1390, 1737, 2171, 2714, 3392, 4240, 5300, 6625, 8281, 10351, 12939, 16174, 20217, 25271, 31589, 39486, 49357, 61696, 77120

Offset: 1

N. J. A. Sloane, Dec 01 2004

nonn

Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. This is Sieve #6 with K=5. [Annotated and scanned copy]
Index entries for sequences generated by sieves

Cf. A003309, A003310, A100464, A100562, A003312, A003311, A052548, A100585.

t = Range[3, 80000]; r = {}; While[Length[t] > 0, AppendTo[r, First[t]]; t = Drop[t, {1, -1, 5}];]; r (* Ray Chandler, Dec 02 2004 *)

A262231 First term is 2; each subsequent term is the least number greater than the previous term but not a multiple of the successor of any previous term.

Original entry on oeis.org

2, 4, 7, 11, 13, 17, 19, 22, 26, 29, 31, 34, 37, 41, 43, 47, 49, 52, 58, 61, 67, 71, 73, 77, 79, 82, 86, 89, 91, 94, 97, 101, 103, 107, 109, 113, 116, 119, 121, 127, 131, 133, 137, 139, 142, 146, 149, 151, 157, 163, 167, 169, 172, 178, 181, 187, 191

Offset: 1

Drake Thomas, Sep 15 2015

nonn

If the twin prime conjecture is true, this sequence has infinitely many pairs of terms with difference at most 3.

			25 is excluded because it is a multiple of 4+1; 26 is not a multiple of 3,5,8,...,23 so it remains in the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A100464.

has(n)=fordiv(n,d, if(mapisdefined(m, d-1), return(0))); 1
first(n)=local(m=Map(Mat([2,0]))); my(t=3); while(#mCharles R Greathouse IV, Sep 16 2015

a(n) = A100464(n) - 1.

cp's OEIS Frontend

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

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A100585 a(n+1) = a(n)+floor(a(n)/3), a(1) = 3.

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A003310 Generated by a sieve.

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A003312 a(1) = 3; for n>0, a(n+1) = a(n) + floor((a(n)-1)/2).

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A003311 Write down the numbers from 3 to infinity. Take next number, M say, that has not been crossed off. Counting through the numbers that have not yet been crossed off after that M, cross off the first, (M+1)st, (2M+1)st, (3M+1)st, etc. Repeat. The numbers that are left form the sequence.

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Haskell

Extensions

A100562 Write down the numbers from 7 to infinity. Take next number, M say, that has not been crossed off and cross off all the numbers i*M - 1 for i >= 2. Repeat. The numbers that are left form the sequence.

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