A376237 -id:A376237 - 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

A255127 Ludic array: square array A(row,col), where row n lists the numbers removed at stage n in the sieve which produces Ludic numbers. Array is read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

2, 4, 3, 6, 9, 5, 8, 15, 19, 7, 10, 21, 35, 31, 11, 12, 27, 49, 59, 55, 13, 14, 33, 65, 85, 103, 73, 17, 16, 39, 79, 113, 151, 133, 101, 23, 18, 45, 95, 137, 203, 197, 187, 145, 25, 20, 51, 109, 163, 251, 263, 281, 271, 167, 29, 22, 57, 125, 191, 299, 325, 367, 403, 311, 205, 37, 24, 63, 139, 217, 343, 385, 461, 523, 457, 371, 253, 41

Offset: 2

Antti Karttunen, Feb 22 2015

The starting offset of the sequence giving the terms of square array is 2. However, we can tacitly assume that a(1) = 1 when the sequence is used as a permutation of natural numbers. However, term 1 itself is out of the array.

The choice of offset = 2 for the terms starting in rows >= 1 is motivated by the desire to have a permutation of the integers n -> a(n) with a(n) = A(A002260(n-1), A004736(n-1)) for n > 1 and a(1) := 1. However, since this sequence is declared as a "table", offset = 2 would mean that the first *row* (not element) has index 2. I think the sequence should have offset = 1 and the permutation of the integers would be n -> a(n-1) with a(0) := 1 (if a(1) = A(1,1) = 2). Or, the sequence could have offset 0, with an additional row 0 of length 1 with the only element a(0) = A(0,1) = 1, the permutation still being n -> a(n-1) if a(n=0, 1, 2, ...) = (1, 2, 4, ...). This would be in line with considering 1 as the first ludic number, and A(n, 1) = A003309(n+1) for n >= 0. - M. F. Hasler, Nov 12 2024

			The top left corner of the array:
   2,   4,   6,   8,  10,  12,   14,   16,   18,   20,   22,   24,   26
   3,   9,  15,  21,  27,  33,   39,   45,   51,   57,   63,   69,   75
   5,  19,  35,  49,  65,  79,   95,  109,  125,  139,  155,  169,  185
   7,  31,  59,  85, 113, 137,  163,  191,  217,  241,  269,  295,  323
  11,  55, 103, 151, 203, 251,  299,  343,  391,  443,  491,  539,  587
  13,  73, 133, 197, 263, 325,  385,  449,  511,  571,  641,  701,  761
  17, 101, 187, 281, 367, 461,  547,  629,  721,  809,  901,  989, 1079
  23, 145, 271, 403, 523, 655,  781,  911, 1037, 1157, 1289, 1417, 1543
  25, 167, 311, 457, 599, 745,  883, 1033, 1181, 1321, 1469, 1615, 1753
  29, 205, 371, 551, 719, 895, 1073, 1243, 1421, 1591, 1771, 1945, 2117
...

Antti Karttunen, Table of n, a(n) for n = 2..10441; the first 144 antidiagonals of array, flattened

Transpose: A255129.
Inverse: A255128. (When considered as a permutation of natural numbers with a(1) = 1).
Cf. A260738 (index of the row where n occurs), A260739 (of the column).
Main diagonal: A255410.
Column 1: A003309 (without the initial 1). Column 2: A254100.
Row 1: A005843, Row 2: A016945, Row 3: A255413, Row 4: A255414, Row 5: A255415, Row 6: A255416, Row 7: A255417, Row 8: A255418, Row 9: A255419.
A192607 gives all the numbers right of the leftmost column, and A192506 gives the composites among them.
Cf. A272565, A271419, A271420 and permutations A269379, A269380, A269384.
Cf. also related or derived arrays A260717, A257257, A257258 (first differences of rows), A276610 (of columns), A276580.
Analogous arrays for other sieves: A083221, A255551, A255543.
Cf. A376237 (ludic factorials), A377469 (ludic analog of A005867).

rows = 12; cols = 12; t = Range[2, 3000]; r = {1}; n = 1; While[n <= rows, k = First[t]; AppendTo[r, k]; t0 = t; t = Drop[t, {1, -1, k}]; ro[n++] = Complement[t0, t][[1 ;; cols]]]; A = Array[ro, rows]; Table[ A[[n - k + 1, k]], {n, 1, rows}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Mar 14 2016, after Ray Chandler *)

a255127 = lambda n: A255127(A002260(k-1), A004736(k-1))
def A255127(n, k):
    A = A255127; R = A.rows
    while len(R) <= n or len(R[n]) < min(k, A.P[n]): A255127_extend(2*n)
    return R[n][(k-1) % A.P[n]] + (k-1)//A.P[n] * A.S[n]
A=A255127; A.rows=[[1],[2],[3]]; A.P=[1]*3; A.S=[0,2,6]; A.limit=30
def A255127_extend(rMax=9, A=A255127):
    A.limit *= 2; L = [x+5-x%2 for x in range(0, A.limit, 3)]
    for r in range(3, rMax):
        if len(A.P) == r:
            A.P += [ A.P[-1] * (A.rows[-1][0] - 1) ]  # A377469
            A.rows += [[]]; A.S += [ A.S[-1] * L[0] ] # ludic factorials
        if len(R := A.rows[r]) < A.P[r]: # append more terms to this row
            R += L[ L[0]*len(R) : A.S[r] : L[0] ]
        L = [x for i, x in enumerate(L) if i%L[0]] # M. F. Hasler, Nov 17 2024

(define (A255127 n) (if (<= n 1) n (A255127bi (A002260 (- n 1)) (A004736 (- n 1)))))
(define (A255127bi row col) ((rowfun_n_for_A255127 row) col))
;; definec-macro memoizes its results:
(definec (rowfun_n_for_A255127 n) (if (= 1 n) (lambda (n) (+ n n)) (let* ((rowfun_for_remaining (rowfun_n_for_remaining_numbers (- n 1))) (eka (rowfun_for_remaining 0))) (COMPOSE rowfun_for_remaining (lambda (n) (* eka (- n 1)))))))
(definec (rowfun_n_for_remaining_numbers n) (if (= 1 n) (lambda (n) (+ n n 3)) (let* ((rowfun_for_prevrow (rowfun_n_for_remaining_numbers (- n 1))) (off (rowfun_for_prevrow 0))) (COMPOSE rowfun_for_prevrow (lambda (n) (+ 1 n (floor->exact (/ n (- off 1)))))))))

From M. F. Hasler, Nov 12 2024: (Start)
A(r, c) = A(r, c-P(r)) + S(r) = A(r, ((c-1) mod P(r)) + 1) + floor((c-1)/P(r))*S(r) with periods P = (1, 1, 2, 8, 48, 480, 5760, ...) = A377469, and shifts S = (2, 6, 30, 210, 2310, 30030, 510510) = A376237(2, 3, ...). For example:
A(1, c) = A(1, c-1) + 2 = 2 + (c-1)*2 = 2*c,
A(2, c) = A(2, c-1) + 6 = 3 + (c-1)*6 = 6*c - 3,
A(3, c) = A(3, c-2) + 30 = {5 if c is odd else 19} + floor((c-1)/2)*30 = 15*c - 11 + (c mod 2),
A(4, c) = A(4, c-8) + 210 = A(4, ((c-1) mod 8)+1) + floor((c-1)/8)*210, etc. (End)

A255418 Row 8 of Ludic array A255127.

Original entry on oeis.org

23, 145, 271, 403, 523, 655, 781, 911, 1037, 1157, 1289, 1417, 1543, 1673, 1801, 1927, 2057, 2183, 2305, 2437, 2563, 2693, 2819, 2951, 3071, 3197, 3331, 3457, 3587, 3713, 3841, 3967, 4093, 4223, 4349, 4477, 4603, 4735, 4855, 4987, 5113, 5237, 5369, 5489, 5621, 5747, 5875, 6001

Offset: 1

Antti Karttunen, Feb 22 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001

Row 8 of A255127. See A255417 for row 7 and A255419 for row 9.

my(L=vector(3913910, x, 3*x+1+x%2), m(n, k)=2^(n\/k*k)\(2^k-1)); for(i=3, 7, L=vecextract(L, 2^#L-m(#L, L[1])-1)); L255418=vecextract(L, m(#L, L[1]));
A255418(n, P=92160)=n--\P*11741730 + L255418[n%P+1] \\ M. F. Hasler, Nov 17 2024

# S can be decreased if only terms up to a smaller limit are needed.
def A255418(n, S=11741730, P=92160):
    try: n-=1; return A255418.L[n]
    except IndexError: return A255418.L[n%P] + n//P*S
    except AttributeError: L = [x+5-x%2 for x in range(0, S, 3)]
    while (k:=L[0]) < 23: L = [x for i, x in enumerate(L) if i%k]
    A255418.L = L[::k]; return A255418(n+1) # M. F. Hasler, Nov 17 2024

(define (A255418 n) (A255127bi 8 n)) ;; Code for A255127bi given in A255127.

a(n) = a(n-P) + S = a((n-1)%P + 1) + S*floor((n-1)/P) with period P = 92160 = A377469(8) and shift S = 11741730 = A376237(9). - M. F. Hasler, Nov 17 2024

A255419 Row 9 of Ludic array A255127.

Original entry on oeis.org

25, 167, 311, 457, 599, 745, 883, 1033, 1181, 1321, 1469, 1615, 1753, 1903, 2041, 2191, 2339, 2483, 2623, 2773, 2911, 3059, 3211, 3353, 3493, 3637, 3781, 3929, 4067, 4217, 4367, 4507, 4657, 4795, 4937, 5087, 5227, 5377, 5527, 5665, 5813, 5957, 6101, 6241, 6389, 6535, 6683, 6821, 6971, 7111

Offset: 1

Antti Karttunen, Feb 22 2015

nonn

This is the first row in A255127 that starts with a composite number.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Row 9 of A255127. See A255413 - A255418 for rows 3 through 8.

my(L=vector(97847750, x, 3*x+1+x%2), m(n, k)=2^(n\/k*k)\(2^k-1)); for(i=3, 8, L=vecextract(L, 2^#L-m(#L, L[1])-1)); L255419=vecextract(L, m(#L, L[1]));
\\ If only terms up to N < P are needed, the vector L above can be chosen shorter
A255419(n, P=2027520)=n--\P*293543250 + L255419[n%P+1] \\ M. F. Hasler, Nov 17 2024

# if only terms up to a smaller limit S are needed, then S can be decreased
def A255419(n, S=293543250, P=2027520):
    try: n -= 1; return A255419.L[n]
    except IndexError: return A255419.L[n%P] + n//P*S
    except AttributeError: L = [x+5-x%2 for x in range(0, S, 3)]
    while (k:=L[0]) < 25: L = [x for i, x in enumerate(L) if i%k]
    A255419.L = L[::k]; return A255419(n+1) # M. F. Hasler, Nov 17 2024

(define (A255419 n) (A255127bi 9 n)) ;; Code for A255127bi given in A255127.

a(n) = a(n-P) + S = a((n-1)%P + 1) + S*floor((n-1)/P) with period P = 2027520 = A377469(9) and shift S = 293543250 = A376237(10). - M. F. Hasler, Nov 17 2024

A377469 a(n) = (A003309(n)-1)*a(n-1) (n > 1), a(1) = 1, where A003309 are the ludic numbers.

Original entry on oeis.org

1, 1, 2, 8, 48, 480, 5760, 92160, 2027520, 48660480, 1362493440, 49049763840, 1961990553600, 82403603251200, 3790565749555200, 197109418976870400, 11826565138612224000, 780553299148406784000, 54638730940388474880000, 4152543551469524090880000, 340508571220500975452160000

Offset: 1

M. F. Hasler, Nov 13 2024

nonn

The analog of A005867 for ludic numbers A003309 instead of primes A000040. Since A003309(9) = 23 > A000040(8) = 19 is the first term that differs in the two sequences (up to offset and initial 1's), this sequence differs from A005867 also from the 9th term on, which is a(9) = (23-1)*92160 = 2027520 instead of A005867(8) = (19-1)*92160 = 1658880.

This sequence gives the (pseudo) period lengths of the rows of the ludic sieve array A255127, in which row r is pseudo-periodic, A255127(r, c) = A255127(r, c-a(r)) + S(r), with the shift S(r) given by the ludic factorials A376237.

			Since A003309 = (1, 2, 3, 5, 7, 11, 13,17, 23, 25, ...), we get:
a(2) = (2-1)*1 = 1, a(3) = (3-1)*1 = 2, a(4) = (5-1)*2 = 8, a(5) = (7-1)*8 = 48,
a(6) = (11-1)*48 = 480, a(7) = (13-1)*480 = 5760, a(8) = (17-1)*2 = 92160,
a(9) = (23-1)*92160 = 2027520, a(10) = (25-1)*2027520 = 48660480, and so on.

Cf. A003309 (ludic numbers), A255127 (ludic sieve array), A376237 (ludic factorials), A005867 (analog of this sequence for primes), A000040 (the primes).

apply( {A377469(n) = if(n>1, (A003309(n)-1)*A377469(n-1), 1)}, [1..19])

a(n) = A005867(n-1) up to n = 8.

A376236 Ludic Fortunate numbers: a(n) = N(P(n)+1) - P(n), where N(x) = min {L in A003309 | L > x} is the next larger ludic number and P(n) = Prod_{k=1..n} A003309[n].

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 17, 37, 61

Offset: 1

M. F. Hasler, Nov 02 2024

Generalization of Fortunate numbers A005235 to ludic numbers A003309 instead of primes.

Are all terms ludic numbers? Will all ludic numbers > 1 appear in this sequence?

			The first ludic numbers are A003309 = 1, 2, 3, 5, 7, 11, 13, 17, 23, 25, 29, 37, ...
Their cumulative products are P = 1, 2, 6, 30, 210, 2310, 30030, 510510, 11741730, ...
Up to 510510 they are the same as primorials A002110 because ludic numbers > 1 coincide with the primes up to 17.
The first term of this sequence is a(1) = N(1 + P(1)) - P(1) = N(2) - 1 = 3 - 1 = 2, where we write N(x) for the least A003309(k) > x.
The second term is a(2) = N(1 + P(2)) - P(2) = N(3) - 2 = 5 - 2 = 3.
Then a(3) = N(1 + P(3)) - P(3) = N(7) - 6 = 11 - 6 = 5.
Then a(4) = N(1 + P(4)) - P(4) = N(31) - 30 = 37 - 30 = 7, still as in A005235 (because that sequence also uses the least strictly larger prime).
Then a(5) = N(1 + P(5)) - P(5) = N(211) - 210 = 221 - 210 = 11 (while A005235 has nextprime(211) - 210 = 223 - 210 = 13, where again it does not matter that 211 is a prime).

Cf. A003309 (ludic numbers), A376237 (ludic factorials), A005235 (Fortunate numbers: same idea with primes).

A376236(n)=next_A003309(1+n=A376237(n))-n

a(9) from Pontus von Brömssen, Nov 03 2024

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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A255127 Ludic array: square array A(row,col), where row n lists the numbers removed at stage n in the sieve which produces Ludic numbers. Array is read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A255418 Row 8 of Ludic array A255127.

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A255419 Row 9 of Ludic array A255127.

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A377469 a(n) = (A003309(n)-1)*a(n-1) (n > 1), a(1) = 1, where A003309 are the ludic numbers.

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A376236 Ludic Fortunate numbers: a(n) = N(P(n)+1) - P(n), where N(x) = min {L in A003309 | L > x} is the next larger ludic number and P(n) = Prod_{k=1..n} A003309[n].

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A372607 Let a(1) = 2, f(n) = a(1)*a(2)*...*a(n-1) for n >= 1 and a(n) = nextludicnumber(f(n)+1) - f(n) for n >= 2, where nextludicnumber(x) is the smallest ludic number > x.

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A372607 Let a(1) = 2, f(n) = a(1)a(2)...*a(n-1) for n >= 1 and a(n) = nextludicnumber(f(n)+1) - f(n) for n >= 2, where nextludicnumber(x) is the smallest ludic number > x.