A235491 -id:A235491 - 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

A237126 a(0)=0, a(1) = 1, a(2n) = nonludic(a(n)), a(2n+1) = ludic(a(n)+1), where ludic = A003309, nonludic = A192607.

Original entry on oeis.org

0, 1, 4, 2, 9, 7, 6, 3, 16, 25, 14, 17, 12, 13, 8, 5, 26, 61, 36, 115, 22, 47, 27, 67, 20, 41, 21, 43, 15, 23, 10, 11, 38, 119, 81, 359, 51, 179, 146, 791, 33, 91, 64, 247, 39, 121, 88, 407, 31, 83, 57, 221, 32, 89, 59, 227, 24, 53, 34, 97, 18, 29, 19, 37, 54

Offset: 0

Antti Karttunen and Reinhard Zumkeller, Feb 06 2014

nonn

Shares with permutation A237056 the property that the other bisection consists of only ludic numbers and the other bisection of only nonludic numbers. However, instead of placing terms in those subsets in monotone order this sequence recursively permutes the order of both subsets with the emerging permutation itself, so this is a kind of "deep" variant of A237056.

Alternatively, this can be viewed as yet another "entanglement permutation", where two pairs of complementary subsets of natural numbers are entangled with each other. In this case a complementary pair odd/even numbers (A005408/A005843) is entangled with a complementary pair ludic/nonludic numbers (A003309/A192607).

			a(2) = a(2*1) = nonludic(a(1)) = A192607(1) = 4.
a(3) = a(2*1+1) = ludic(a(1)+1) = A003309(1+1) = A003309(2) = 2.
a(4) = a(2*2) = nonludic(a(2)) = A192607(4) = 9.
a(5) = a(2*2+1) = ludic(a(2)+1) = A003309(4+1) = A003309(5) = 7.

Antti Karttunen, Table of n, a(n) for n = 0..574
Index entries for sequences that are permutations of the natural numbers

Cf. A237427 (inverse), A237056, A235491.

Similarly constructed permutations: A227413/A135141.

import Data.List (transpose)
a237126 n = a237126_list !! n
a237126_list = 0 : es where
   es = 1 : concat (transpose [map a192607 es, map (a003309 . (+ 1)) es])
-- Reinhard Zumkeller, Feb 10 2014, Feb 06 2014

nmax = 64;
T = Range[2, 20 nmax];
L = {1};
While[Length[T] > 0, With[{k = First[T]},
     AppendTo[L, k]; T = Drop[T, {1, -1, k}]]];
nonL = Complement[Range[Last[L]], L];
a[n_] := a[n] = Which[
     n < 2, n,
     EvenQ[n] && a[n/2] <= Length[nonL], nonL[[a[n/2]]],
     OddQ[n] && a[(n-1)/2]+1 <= Length[L], L[[a[(n-1)/2]+1]],
     True, Print[" error: n = ", n, " size of T should be increased"]];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Oct 10 2021, after Ray Chandler in A003309 *)

;; With Antti Karttunen's IntSeq-library for memoizing definec-macro.
(definec (A237126 n) (cond ((< n 2) n) ((even? n) (A192607 (A237126 (/ n 2)))) (else (A003309 (+ 1 (A237126 (/ (- n 1) 2))))))) ;; Antti Karttunen, Feb 07 2014

a(0)=0, a(1) = 1, a(2n) = nonludic(a(n)), a(2n+1) = ludic(a(n)+1), where ludic = A003309, nonludic = A192607.

A244319 Self-inverse permutation of natural numbers: a(1) = 1, a(2n) = A003961(1+a(A064989(2n-1))), a(2n+1) = 1+A003961(a(A064989(2n+1)-1)).

Original entry on oeis.org

1, 3, 2, 9, 6, 5, 26, 11, 4, 21, 8, 125, 56, 25, 16, 15, 344, 115, 36, 1015, 10, 39, 204, 41, 14, 7, 52, 45, 86, 301, 176, 155, 298, 51, 50, 19, 518, 305, 22, 189, 24, 895, 1376, 49, 28, 825, 1268, 11875, 44, 35, 34, 27, 3186, 6625, 2388, 13, 454, 153, 126, 3191, 476, 131

Offset: 1

Antti Karttunen, Jul 18 2014; description corrected and PARI code added Jul 30 2014

nonn

After 1, maps each even number to a unique odd number and vice versa, i.e., for all n > 1, A000035(a(n)) XOR A000035(n) = 1, where XOR is given in A003987.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A003987, A003961, A064989, A243501.
Related permutations: A048673, A064216, A245609-A245610.
Similar entanglement permutations: A245605-A245606, A235491, A236854, A243347, A244152.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ This function from Michel Marcus
A064989(n) = my(f = factor(n)); for(i=1, #f~, if((2 == f[i,1]),f[i,1] = 1,f[i,1] = precprime(f[i,1]-1))); factorback(f);
A244319(n) = if(1==n, 1, if(0==(n%2), A003961(1+A244319(A064989(n-1))), 1+A003961(A244319(A064989(n)-1))));
for(n=1, 10001, write("b244319.txt", n, " ", A244319(n)))
(Scheme, with Antti Karttunen's IntSeq-library for memoizing definec-macro)
(definec (A244319 n) (cond ((= 1 n) 1) ((even? n) (A003961 (+ 1 (A244319 (A064989 (- n 1)))))) (else (A243501 (A244319 (-1+ (A064989 n)))))))

a(1) = 1, a(2n) = A003961(1+a(A064989(2n-1))), a(2n+1) = A243501(a(A064989(2n+1)-1)).
As a composition of related permutations:
a(n) = A245609(A048673(n)) = A064216(A245610(n)).

A243347 a(1)=1, and for n>1, if mu(n) = 0, a(n) = A005117(1+a(A057627(n))), otherwise, a(n) = A013929(a(A013928(n))).

Original entry on oeis.org

1, 4, 12, 2, 32, 8, 84, 6, 19, 24, 220, 3, 18, 50, 63, 53, 564, 13, 9, 138, 49, 128, 162, 10, 31, 136, 38, 365, 1448, 36, 25, 5, 351, 126, 332, 30, 414, 27, 81, 82, 348, 99, 931, 103, 86, 3699, 96, 929, 21, 14, 64, 223, 16, 79, 892, 210, 325, 847, 80, 265, 1056, 72, 15, 51, 208, 212, 884, 221, 256

Offset: 1

Antti Karttunen, Jun 03 2014

nonn

Self-inverse permutation of natural numbers.

Shares with A088609 the property that after 1, positions indexed by squarefree numbers larger than one, A005117(n+1): 2, 3, 5, 6, 7, 10, 11, 13, 14, ... contain only nonsquarefree numbers A013929: 4, 8, 9, 12, 16, 18, 20, 24, ..., and vice versa. However, instead of placing terms in those subsets in monotone order this sequence recursively permutes the order of both subsets with the emerging permutation itself, thus implementing a kind of "deep" variant of A088609. Alternatively, this can be viewed as yet another "entanglement permutation", where two pairs of complementary subsets of natural numbers are interwoven with each other. In this case complementary pair A005117/A013929 is entangled with complementary pair A013929/A005117.

Antti Karttunen, Table of n, a(n) for n = 1..520
Index entries for sequences that are permutations of the natural numbers

Cf. A008966, A005117, A013929, A013928, A057627, A088609, A243348.

Similar permutations: A236854, A235491, A243343-A243346, A243347, A243287-A243288, A135141-A227413, A237126-A237427, A193231.

a(1), and for n>1, if mu(n) = 0, a(n) = A005117(1+a(A057627(n))), otherwise, a(n) = A013929(a(A013928(n))). [Here mu is Moebius mu-function, A008683, which is zero only when n is a nonsquarefree number, one of the numbers in A013929.]

For all n > 1, A008966(a(n)) = 1 - A008966(n), or equally, mu(a(n)) + 1 = mu(n) modulo 2, where mu is Moebius mu (A008683). [Note: Permutation A088609 satisfies the same condition.]

A244152 Self-inverse permutation of natural numbers: a(1) = 1; thereafter, if n is k-th number with an odd number of prime divisors (counted with multiplicity) [i.e., n = A026424(k)], a(n) = A028260(1+a(k)), otherwise, when n is k-th number > 1 with an even number of prime divisors [i.e., n = A028260(1+k)], a(n) = A026424(a(k)).

Original entry on oeis.org

1, 4, 10, 2, 24, 7, 6, 55, 18, 3, 16, 15, 121, 44, 12, 11, 39, 9, 36, 35, 105, 31, 250, 5, 29, 28, 93, 26, 25, 86, 22, 82, 238, 79, 20, 19, 81, 72, 17, 68, 218, 65, 517, 14, 62, 67, 60, 202, 195, 57, 59, 56, 185, 477, 8, 52, 50, 175, 51, 47, 177, 45, 495, 167, 42, 161, 46, 40, 162, 169, 150, 38, 143, 455, 459, 140, 153, 1060, 34, 134, 37, 32

Offset: 1

Antti Karttunen, Jul 27 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Cf. A008836, A026424, A028260, A055037, A055038, A066829.

Similar entanglement permutations: A245603-A245604, A235491, A236854, A243347, A244319.

a(1) = 1, and for n > 1, if A066829(n) = 1, then a(n) = A028260(1 + A244152(A055038(n))), otherwise a(n) = A026424(A244152(A055037(n)-1)).

For all n > 1, A008836(a(n)) = -1 * A008836(n), where A008836 is Liouville's lambda-function.

A245812 Self-inverse permutation of natural numbers: a(0) = 0, a(1) = 1, and for n > 1, if A065620(n) < 0, a(n) = A065621(1+a(-(A065620(n)))), otherwise a(n) = A048724(a(A065620(n)-1)).

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 4, 5, 15, 14, 13, 12, 11, 10, 9, 8, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 57, 56, 59, 58, 61, 60, 63, 62, 49, 48, 51, 50, 53, 52, 55, 54, 41, 40, 43, 42, 45, 44, 47, 46, 33, 32, 35, 34, 37, 36, 39, 38, 106, 107, 104, 105, 110, 111, 108, 109, 98, 99, 96, 97, 102, 103, 100

Offset: 0

Antti Karttunen, Aug 20 2014

nonn

This is an instance of entanglement permutation, where complementary pair A048724/A065621 is entangled with the same pair in the opposite order: A065621/A048724, with a(1) set to 1.

Note how this is A193231-conjugate of A054429.

Similar or related permutations: A193231, A234025, A234026, A246211, A236854, A235491, A054429, A234027.

Cf. A010060, A048724, A065620, A065621, A246159, A246160.

a048724(n) = bitxor(n, 2*n);
a065620(n) = if(n<3, n, if(n%2, -2*a065620((n - 1)/2) + 1, 2*a065620(n/2)));
a065621(n) = bitxor(n, 2*(n - bitand(n, -n)));
a(n) = x=a065620(n); if(n<2, n, if(x<0, a065621(1 + a(-x)), a048724(a(x - 1))));
for(n=0, 100, print1(a(n),", ")) \\ Indranil Ghosh, Jun 07 2017

def a048724(n): return n^(2*n)
def a065620(n): return n if n<3 else 2*a065620(n//2) if n%2==0 else -2*a065620((n - 1)//2) + 1
def a065621(n): return n^(2*(n - (n & -n)))
def a(n):
    x=a065620(n)
    return n if n<2 else a065621(1 + a(-x)) if x<0 else a048724(a(x - 1))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 07 2017

a(0) = 0, a(1) = 1, and for n > 1, if A065620(n) < 0, a(n) = A065621(1+a(-(A065620(n)))), otherwise a(n) = A048724(a(A065620(n)-1)).
Equally:
a(0) = 0, a(1) = 1, and for n > 1, if A010060(n) = 0, a(n) = A065621(1+a(A246159(n))), otherwise a(n) = A048724(a(A246160(n)-1)). [Note how A246159 is an inverse function for A048724, while A246160 is an inverse function for A065621].
As a composition of related permutations:
a(n) = A193231(A234025(n)).
a(n) = A234026(A193231(n)).
a(n) = A193231(A054429(A193231(n))).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

Python

Python

Scheme

Formula

Extensions

A237126 a(0)=0, a(1) = 1, a(2n) = nonludic(a(n)), a(2n+1) = ludic(a(n)+1), where ludic = A003309, nonludic = A192607.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Scheme

Formula

A244319 Self-inverse permutation of natural numbers: a(1) = 1, a(2n) = A003961(1+a(A064989(2n-1))), a(2n+1) = 1+A003961(a(A064989(2n+1)-1)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A243347 a(1)=1, and for n>1, if mu(n) = 0, a(n) = A005117(1+a(A057627(n))), otherwise, a(n) = A013929(a(A013928(n))).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Views

Author

Keywords

Links

Crossrefs

Formula

A245812 Self-inverse permutation of natural numbers: a(0) = 0, a(1) = 1, and for n > 1, if A065620(n) < 0, a(n) = A065621(1+a(-(A065620(n)))), otherwise a(n) = A048724(a(A065620(n)-1)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Python

Formula