A255127 -id:A255127 - cp's OEIS frontend

A255416 Row 6 of Ludic array A255127.

13, 73, 133, 197, 263, 325, 385, 449, 511, 571, 641, 701, 761, 823, 887, 947, 1013, 1075, 1139, 1199, 1261, 1327, 1387, 1447, 1513, 1573, 1637, 1703, 1763, 1825, 1889, 1951, 2011, 2071, 2141, 2201, 2261, 2327, 2387, 2453, 2515, 2575, 2639, 2699, 2767, 2827, 2887, 2953, 3013, 3073, 3143, 3203, 3265, 3325, 3389, 3451, 3511, 3581, 3641, 3701

Offset: 1

Antti Karttunen, Feb 22 2015

nonn

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

Row 6 of A255127. See A255415 for row 5 and A255417 for row 7.

Cf. A084970, A255407.

my(L=[x+2^(x%2)|x<-[1..10^4]*3], m(n,k)=2^(n\/k*k)\(2^k-1)); for(i=3, 5, L=vecextract(L, 2^#L-m(#L, L[1])-1)); L255416=vecextract(L, m(#L, L[1]));
A255416(n)=n--\480*30030+L255416[n%480+1] \\ M. F. Hasler, Nov 17 2024

def A255416(n):
    try: n-=1; return A255416.L[n]
    except IndexError: return n//480*30030 + A255416.L[n%480]
    except AttributeError: L = [3*x+5-(x&1) for x in range(10**4)]
    for k in L[:3]: L = [x for i,x in enumerate(L) if i%k]
    A255416.L = L[::13]; return n//480*30030 + A255416.L[n%480]
# M. F. Hasler, Nov 17 2024

(define (A255416 n) (A255127bi 6 n)) ;; Code for A255127bi given in A255127.

a(n) = A255407(A084970(n)).

a(n) = a(n-480) + 30030 = 30030*floor((n-1)/480) + a((n-1)%480 + 1), where % is the modulo or remainder operator. - M. F. Hasler, Nov 10 2024 and Nov 17 2024

A255417 Row 7 of Ludic array A255127.

Original entry on oeis.org

17, 101, 187, 281, 367, 461, 547, 629, 721, 809, 901, 989, 1079, 1169, 1253, 1343, 1433, 1523, 1613, 1697, 1783, 1873, 1963, 2051, 2143, 2231, 2311, 2407, 2491, 2587, 2671, 2759, 2851, 2935, 3029, 3115, 3205, 3295, 3383, 3467, 3557, 3647, 3737, 3823, 3913, 4001, 4091, 4177, 4271, 4357, 4447, 4537, 4619, 4711, 4799, 4885, 4979, 5065, 5155, 5243

Offset: 1

Antti Karttunen, Feb 22 2015

nonn

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

Row 7 of A255127. See A255416 for row 6 and A255418 for row 8.

my(L=vector(170170,x,3*x+2^(x%2)), m(n, k)=2^(n\/k*k)\(2^k-1)); for(i=3, 6, L=vecextract(L, 2^#L-m(#L, L[1])-1)); L255417=vecextract(L, m(#L, L[1]));
A255417(n)=n--\5760*510510+L255417[n%5760+1] \\ M. F. Hasler, Nov 17 2024

def A255417(n, S=510510, P=5760):
    try: n -= 1; return A255417.L[n]
    except IndexError: return A255417.L[n%P] + n//P*S
    except AttributeError: L = [x+5-x%2 for x in range(0, S, 3)]
    for k in L[:4]: L = [x for i,x in enumerate(L) if i%k]
    A255417.L = L[::17]; return A255417(n+1) # M. F. Hasler, Nov 17 2024

(define (A255417 n) (A255127bi 7 n)) ;; Code for A255127bi given in A255127.

a(n) = a(n-5760) + 510510 = a((n-1)%5760 + 1) + 510510*floor((n-1)/5760). - M. F. Hasler, Nov 10 2024 and Nov 17 2024

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

A257258 Square array A(row,col) = (1/2) * (A255127(row,col+1) - A255127(row,col)): half of the first differences of each row of Ludic array.

Original entry on oeis.org

1, 1, 3, 1, 3, 7, 1, 3, 8, 12, 1, 3, 7, 14, 22, 1, 3, 8, 13, 24, 30, 1, 3, 7, 14, 24, 30, 42, 1, 3, 8, 12, 26, 32, 43, 61, 1, 3, 7, 13, 24, 33, 47, 63, 71, 1, 3, 8, 14, 24, 31, 43, 66, 72, 88, 1, 3, 7, 13, 22, 30, 47, 60, 73, 83, 108, 1, 3, 8, 12, 24, 32, 43, 66, 71, 90, 117, 126

Offset: 1

Antti Karttunen, Apr 29 2015

The array A(row,col) is read by downwards antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

			The top left corner of the array:
    1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1
    3,   3,   3,   3,   3,   3,   3,   3,   3,   3,   3,   3,   3,   3,   3,   3
    7,   8,   7,   8,   7,   8,   7,   8,   7,   8,   7,   8,   7,   8,   7,   8
   12,  14,  13,  14,  12,  13,  14,  13,  12,  14,  13,  14,  12,  13,  14,  13
   22,  24,  24,  26,  24,  24,  22,  24,  26,  24,  24,  24,  22,  26,  24,  24
   30,  30,  32,  33,  31,  30,  32,  31,  30,  35,  30,  30,  31,  32,  30,  33
   42,  43,  47,  43,  47,  43,  41,  46,  44,  46,  44,  45,  45,  42,  45,  45
   61,  63,  66,  60,  66,  63,  65,  63,  60,  66,  64,  63,  65,  64,  63,  65
   71,  72,  73,  71,  73,  69,  75,  74,  70,  74,  73,  69,  75,  69,  75,  74
   88,  83,  90,  84,  88,  89,  85,  89,  85,  90,  87,  86,  88,  89,  88,  87
  108, 117, 113, 121, 114, 113, 120, 109, 117, 123, 110, 115, 117, 113, 117, 118
  126, 135, 127, 137, 129, 127, 129, 138, 131, 133, 129, 128, 132, 138, 132, 132
  137, 142, 134, 142, 152, 135, 141, 139, 147, 141, 141, 138, 141, 144, 146, 138
  154, 158, 157, 158, 160, 158, 156, 154, 162, 168, 158, 151, 158, 157, 161, 157
  180, 180, 177, 184, 180, 186, 185, 184, 176, 180, 190, 177, 185, 190, 176, 184
  206, 217, 212, 210, 213, 220, 213, 210, 216, 212, 211, 222, 212, 211, 215, 213
  ...

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of the array

Column 1: A256483.

Cf. A255127, A257257 (same array but with terms multiplied by 2).

(define (A257258 n) (A257258bi (A002260 n) (A004736 n)))
(define (A257258bi row col) (* (/ 1 2) (- (A255127bi row (+ 1 col)) (A255127bi row col)))) ;; Code for A255127bi given in A255127.

A(row,col) = (1/2) * (A255127(row,col+1) - A255127(row,col)).

A(row,col) = A257257(row,col)/2.

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

Original entry on oeis.org

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

A255543 Unlucky array: Row n consists of unlucky numbers removed at the stage n of Lucky sieve.

Original entry on oeis.org

2, 4, 5, 6, 11, 19, 8, 17, 39, 27, 10, 23, 61, 57, 45, 12, 29, 81, 91, 97, 55, 14, 35, 103, 121, 147, 117, 85, 16, 41, 123, 153, 199, 181, 177, 109, 18, 47, 145, 183, 253, 243, 277, 225, 139, 20, 53, 165, 217, 301, 315, 369, 345, 295, 157, 22, 59, 187, 247, 351, 379, 471, 465, 447, 325, 175, 24, 65, 207, 279, 403, 441, 567, 589, 603, 493, 381, 213

Offset: 1

Antti Karttunen, Feb 25 2015

The array A(row,col) is read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

			Top left corner of the square array:
    2,   4,   6,   8,  10,  12,   14,   16,   18,   20,  22,    24,   26,   28,   30
    5,  11,  17,  23,  29,  35,   41,   47,   53,   59,  65,    71,   77,   83,   89
   19,  39,  61,  81, 103, 123,  145,  165,  187,  207, 229,   249,  271,  291,  313
   27,  57,  91, 121, 153, 183,  217,  247,  279,  309, 343,   373,  405,  435,  469
   45,  97, 147, 199, 253, 301,  351,  403,  453,  507, 555,   609,  661,  709,  763
   55, 117, 181, 243, 315, 379,  441,  505,  571,  633, 697,   759,  825,  889,  951
   85, 177, 277, 369, 471, 567,  663,  757,  853,  949, 1045, 1141, 1239, 1333, 1431
  109, 225, 345, 465, 589, 705,  829,  945, 1063, 1185, 1305, 1423, 1549, 1669, 1789
  139, 295, 447, 603, 765, 913, 1075, 1227, 1377, 1537, 1689, 1843, 1999, 2155, 2313
  157, 325, 493, 667, 835, 999, 1177, 1347, 1513, 1687, 1861, 2029, 2205, 2367, 2535
...

Antti Karttunen, Table of n, a(n) for n = 1..5886; the first 108 antidiagonals of the array, flattened

Permutation of A050505.
Row 1: A005843 (after zero), Row 2: A016969.
Column 1: A219178.
Main diagonal: A255549. The first subdiagonal: A255550 (apart from the initial term).
Transpose: A255544.
This is array A255545 without its leftmost column, A000959.
Cf. also arrays A255127 and A255551.

rows = cols = 12; L = 2 Range[0, 2000] + 1; A = Join[{2 Range[cols]}, Reap[For[n = 2, n <= rows, r = L[[n++]]; L0 = L; L = ReplacePart[L, Table[r i -> Nothing, {i, 1, Length[L]/r}]]; Sow[Complement[L0, L][[1 ;; cols]]]]][[2, 1]]]; Table[A[[n - k + 1, k]], {n, 1, Min[rows, cols]}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Mar 15 2016 *)

(define (A255543 n) (A255543bi (A002260 n) (A004736 n)))
(define (A255543bi row col) ((rowfun_n_for_A255543 row) col))
;; Uses the memoizing definec-macro:
(definec (rowfun_n_for_A255543 n) (if (= 1 n) (lambda (n) (+ n n)) (let* ((rowfun_for_remaining (rowfun_n_for_A000959sieve (- n 1))) (eka (A000959 n))) (compose rowfun_for_remaining (lambda (n) (* eka n))))))
(definec (rowfun_n_for_A000959sieve n) (if (= 1 n) A005408shifted (let* ((prevrowfun (rowfun_n_for_A000959sieve (- n 1))) (everynth (prevrowfun n))) (compose-funs prevrowfun (nonzero-pos 1 1 (lambda (i) (modulo i everynth)))))))
(definec (A000959 n) ((rowfun_n_for_A000959sieve n) n))
(define (A005408shifted n) (- (* 2 n) 1))

A255551 Lucky / Unlucky array, shifted version, 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, 5, 7, 8, 11, 19, 9, 10, 17, 39, 27, 13, 12, 23, 61, 57, 45, 15, 14, 29, 81, 91, 97, 55, 21, 16, 35, 103, 121, 147, 117, 85, 25, 18, 41, 123, 153, 199, 181, 177, 109, 31, 20, 47, 145, 183, 253, 243, 277, 225, 139, 33, 22, 53, 165, 217, 301, 315, 369, 345, 295, 157, 37, 24, 59, 187, 247, 351, 379, 471, 465, 447, 325, 175, 43

Offset: 2

Antti Karttunen, Feb 26 2015

Note how in comparison to A255545, the even numbers on the first row have been shifted one step left, "pushing" term 1 out of the array proper. This was done to obtain a better alignment with arrays like A083221 and A255127 associated with other sieves, from which one may then induce permutations like A255553 by cross-referencing.

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

			The top left corner of the array:
   2,   4,   6,   8,  10,  12,  14,   16,   18,   20,   22,   24,   26,   28,   30
   3,   5,  11,  17,  23,  29,  35,   41,   47,   53,   59,   65,   71,   77,   83
   7,  19,  39,  61,  81, 103, 123,  145,  165,  187,  207,  229,  249,  271,  291
   9,  27,  57,  91, 121, 153, 183,  217,  247,  279,  309,  343,  373,  405,  435
  13,  45,  97, 147, 199, 253, 301,  351,  403,  453,  507,  555,  609,  661,  709
  15,  55, 117, 181, 243, 315, 379,  441,  505,  571,  633,  697,  759,  825,  889
  21,  85, 177, 277, 369, 471, 567,  663,  757,  853,  949, 1045, 1141, 1239, 1333
  25, 109, 225, 345, 465, 589, 705,  829,  945, 1063, 1185, 1305, 1423, 1549, 1669
  31, 139, 295, 447, 603, 765, 913, 1075, 1227, 1377, 1537, 1689, 1843, 1999, 2155
  33, 157, 325, 493, 667, 835, 999, 1177, 1347, 1513, 1687, 1861, 2029, 2205, 2367
...

Antti Karttunen, Table of n, a(n) for n = 2..5996; the first 109 antidiagonals of the array, flattened

Inverse: A255552.
Variant of array A255545. (See also A255543).
Row 1: A005843 (even numbers).
Column 1: 2 followed by A000959(2..) (Lucky numbers from their second term onward).
Main diagonal: A255550.
Similar arrays: A083221, A255127.
Associated permutations: A255553, A255554.

(define (A255551 n) (if (<= n 1) n (A255551bi (A002260 (- n 1)) (A004736 (- n 1)))))
(define (A255551bi row col) (cond ((= 1 row) (+ col col)) ((= 1 col) (A000959 row)) (else (A255543bi row (- col 1)))))
;; Other code as in A255543.

For row = 1, A(row,col) = 2*col; For row > 1 and col = 1, A(row,col) = A000959(row); otherwise, A(row,col) = A255543(row,col-1).

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

Max Barrentine, May 09 2016

nonn

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

Max Barrentine, Table of n, a(n) for n = 1..10000
OEIS Wiki, Ludic numbers.
Index entries for sequences generated by sieves

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).

apply( {A272565(n)=A003309(A260738(n)+1)}, [1..99]) \\ M. F. Hasler, Nov 03 2024

(define (A272565 n) (A003309 (+ 1 (A260738 n)))) ;; Antti Karttunen, Sep 11 2016

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)

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

A269380 a(1) = 1, after which, for odd numbers: a(n) = A260739(n)-th number k for which A260738(k) = A260738(n)-1, and for even numbers: a(n) = a(n/2).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 11, 5, 6, 1, 13, 4, 9, 3, 8, 7, 17, 2, 23, 11, 10, 5, 25, 6, 19, 1, 12, 13, 15, 4, 29, 9, 14, 3, 37, 8, 41, 7, 16, 17, 43, 2, 21, 23, 18, 11, 47, 10, 31, 5, 20, 25, 35, 6, 53, 19, 22, 1, 27, 12, 61, 13, 24, 15, 67, 4, 55, 29, 26, 9, 71, 14, 33, 3, 28, 37, 77, 8, 49, 41, 30, 7, 83, 16, 89, 17, 32, 43, 39, 2

Offset: 1

Antti Karttunen, Mar 01 2016

nonn

For odd numbers n > 1, a(n) tells which term is on the immediately preceding row of A255127 (square array generated by Ludic sieve), in the same column where n itself is.

Antti Karttunen, Table of n, a(n) for n = 1..33011

Cf. A255127, A260738, A260739.
Cf. A269172, A269355, A269357, A269382, A269386, A269388 (sequences that use this function).
Cf. also A268674, A269370.

a(1) = 1; after which, for even numbers a(n) = a(n/2), and for odd numbers a(n) = A255127(A260738(n)-1, A260739(n)).
Other identities. For all n >= 1:
a(A269379(n)) = n.

cp's OEIS Frontend

A255416 Row 6 of Ludic array A255127.

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A255417 Row 7 of Ludic array A255127.

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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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A257258 Square array A(row,col) = (1/2) * (A255127(row,col+1) - A255127(row,col)): half of the first differences of each row of Ludic array.

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

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A255543 Unlucky array: Row n consists of unlucky numbers removed at the stage n of Lucky sieve.

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