A272565 -id:A272565 - cp's OEIS frontend

A302034 A028234 analog for a factorization process based on the Ludic sieve (A255127); Discard all instances of the (smallest) Ludic factor A272565(n) from n.

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 7, 5, 1, 1, 9, 1, 5, 1, 11, 1, 3, 1, 13, 7, 7, 1, 15, 1, 1, 5, 17, 7, 9, 1, 19, 11, 5, 1, 21, 1, 11, 1, 23, 1, 3, 1, 25, 19, 13, 1, 27, 1, 7, 7, 29, 11, 15, 1, 31, 13, 1, 11, 33, 1, 17, 5, 35, 1, 9, 1, 37, 17, 19, 1, 39, 7, 5, 11, 41, 1, 21, 1, 43, 35, 11, 1, 45, 1, 23, 1, 47, 13, 3, 1, 49, 23, 25, 1, 51, 13, 13, 19

Offset: 1

Antti Karttunen, Apr 01 2018

nonn

Iterating n, a(n), a(a(n)), a(a(a(n))), ..., until 1 is reached, and taking the Ludic factor (A272565) of each term gives a sequence of distinct Ludic numbers (A003309) in ascending order, while applying A302035 to the same terms gives the corresponding "exponents" of these Ludic factors in this nonstandard "Ludic factorization of n", unique for each natural number n >= 1. Permutation pair A302025/A302026 maps between this Ludic factorization and the ordinary prime factorization of n. See also comments and examples in A302032.

Antti Karttunen, Table of n, a(n) for n = 1..32768
Index entries for sequences generated by sieves

Cf. A003309, A255127, A260738, A260739, A269379, A269380, A302025, A302026, A302032, A302035.
Cf. A302036 (gives the positions of 1's).
Cf. also A028234, A302044.

\\ Assuming A269379 and its inverse A269380 have been precomputed, then the following is reasonably fast:
A302034(n) = if(1==n,n,my(k=0); while((n%2), n = A269380(n); k++); n = (n/2^valuation(n, 2)); while(k>0, n = A269379(n); k--); (n));

For n > 1, a(n) = A269379^(r)(A000265(A260739(n))), where r = A260738(n)-1 and A269379^(r)(n) stands for applying r times the map x -> A269379(x), starting from x = n.

a(n) = A302025(A028234(A302026(n))).

A276347 Numbers n for which A020639(n) = A272565(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 78, 80, 81, 82, 83, 84, 86, 87, 88, 89, 90, 92, 93, 94, 95

Offset: 1

Antti Karttunen, Sep 11 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences generated by sieves

Cf. A020639, A272565.
Complement: A276437.
Subsequence of A276568.
Subsequences: A047229 (after the initial zero), A192503.

A276568 Numbers such that ludic factor of n (A272565) divides n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 86, 87, 88, 89, 90, 91

Offset: 1

Antti Karttunen, Sep 11 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Complement: A276569.
Indices of zeros in A276570.
Cf. A272565, A276440.
Cf. A276347 (subsequence).

A276569 Numbers such that ludic factor of n (A272565) does not divide n.

Original entry on oeis.org

19, 31, 49, 59, 73, 79, 85, 101, 103, 109, 113, 133, 137, 139, 145, 151, 163, 167, 169, 191, 197, 199, 203, 205, 229, 241, 251, 253, 259, 263, 269, 271, 281, 289, 293, 295, 299, 311, 317, 319, 323, 343, 347, 349, 355, 367, 371, 373, 379, 385, 391, 401, 403, 409, 413, 439, 443, 449, 451, 457, 461, 469, 473, 479, 487, 491, 499

Offset: 1

Antti Karttunen, Sep 11 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..3200

Complement: A276568.
Cf. A272565, A276440.
Subsequence of A276437.
Cf. A192505 (a subsequence).

A276437 Numbers n for which A020639(n) <> A272565(n).

Original entry on oeis.org

19, 25, 31, 49, 55, 59, 73, 77, 79, 85, 91, 101, 103, 109, 113, 115, 119, 121, 133, 137, 139, 143, 145, 151, 161, 163, 167, 169, 175, 187, 191, 197, 199, 203, 205, 209, 221, 229, 235, 241, 247, 251, 253, 259, 263, 265, 269, 271, 281, 287, 289, 293, 295, 299, 301, 311, 317, 319, 323, 325, 329, 341, 343, 347, 349, 355, 361

Offset: 1

Antti Karttunen, Sep 11 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..3600
Index entries for sequences generated by sieves

Cf. A020639, A272565.
Complement: A276347.
Disjoint union of A276447 and A276448.
Cf. A276569 (a subsequence).

A302035 a(1) = 0, for n > 1, a(n) = A001511(A260739(n)); Number of instances of (the smallest) Ludic factor A272565(n) in n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 3, 1, 1, 3, 1, 1, 1, 2, 1, 1, 2, 5, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 4, 1, 1, 4, 3, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 3, 1, 1, 3, 2, 1, 1, 2, 1, 1, 2, 4, 2, 1, 1, 2, 3, 1, 1, 3, 1, 1, 1, 2, 5, 1, 1, 5, 1, 1, 1, 2, 2, 1, 1, 3, 2

Offset: 1

Antti Karttunen, Apr 01 2018

nonn

An A067029 analog for "Ludic factorization": iterating the map n -> A302034(n) until 1 is reached, and taking the Ludic factor (A272565) of each term gives a sequence of distinct Ludic numbers (A003309) in ascending order, while applying this function (A302035) to those terms gives the corresponding "exponents" of those Ludic factors, that is, the count of consecutive occurrences of each when iterating the map n -> A302032(n), which gives the same factors with repetitions. Permutation pair A302025/A302026 maps between the Ludic factorization and the ordinary prime factorization of n. See also comments and examples in A302032.

Antti Karttunen, Table of n, a(n) for n = 1..10105
Index entries for sequences generated by sieves

Cf. A001511, A003309, A255127, A260739, A302025, A302026, A302032, A302034.

Cf. also A067029, A302045.

a(1) = 0; for n > 1, a(n) = A001511(A260739(n)).

For n > 1, a(n) = A302025(A067029(A302026(n))).

A276447 Numbers n for which A272565(n) < A020639(n).

Original entry on oeis.org

19, 31, 49, 59, 73, 79, 101, 103, 109, 113, 137, 139, 151, 163, 167, 169, 191, 197, 199, 229, 241, 251, 259, 263, 269, 271, 281, 289, 293, 299, 311, 317, 319, 323, 347, 349, 367, 373, 379, 391, 401, 409, 439, 443, 449, 451, 457, 461, 469, 479, 487, 491, 499, 521, 523, 529, 533, 547, 557, 559, 563, 569, 571, 583, 587, 589, 599, 601

Offset: 1

Antti Karttunen, Sep 11 2016

nonn

			19 is present as A272565(19)=5 and 5 < A020639(19)=19. (19 is right after 5 on the third row of array A255127 while on A083221 it occurs at the beginning of row 8 that starts with 19 itself).
49 is present as it occurs as the fourth number on the third row of A255127 beginning with 5: 5,  19,  35,  49, ..., thus A272565(49)=5, while in A083221 49 occurs right after 7 on row 4, thus A020639(49)=7, and 5 < 7.

Antti Karttunen, Table of n, a(n) for n = 1..2000

Cf. A276448 (complement in A276437), A276347.

Cf. A020639, A272565, A192505, A083221, A255127.

A276448 Numbers n for which A272565(n) > A020639(n).

Original entry on oeis.org

25, 55, 77, 85, 91, 115, 119, 121, 133, 143, 145, 161, 175, 187, 203, 205, 209, 221, 235, 247, 253, 265, 287, 295, 301, 325, 329, 341, 343, 355, 361, 371, 377, 385, 403, 407, 413, 415, 437, 445, 473, 475, 481, 493, 497, 505, 511, 517, 527, 535, 539, 551, 553, 565, 581, 595, 623, 625, 655, 667, 671, 685, 697, 703, 707, 713, 715, 721

Offset: 1

Antti Karttunen, Sep 11 2016

nonn

			85 is the fourth number of the fourth row of A255127, which starts with 7: 7, 31, 59, 85, ..., thus A272565(85)=7, while on A083221 it occurs as the sixth term on the third row that starts with 5, thus A020639(85)=5, and 7 > 5, thus 85 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..2000

Cf. A276447 (complement in A276437), A276347.

Cf. A020639, A272565, A192504, A083221, A255127.

A276570 a(n) = n modulo ludic factor of n: n mod A272565(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 4, 0, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Antti Karttunen, Sep 13 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8008
Index entries for sequences generated by sieves

Cf. A003309, A272565, A276580 (computed for each term of array A255127).

Cf. A276568 (positions of zeros), A276569 (of nonzeros).

(define (A276570 n) (modulo n (A272565 n)))

a(n) = n mod A272565(n).

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

cp's OEIS Frontend

A302034 A028234 analog for a factorization process based on the Ludic sieve (A255127); Discard all instances of the (smallest) Ludic factor A272565(n) from n.

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A276347 Numbers n for which A020639(n) = A272565(n).

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A276568 Numbers such that ludic factor of n (A272565) divides n.

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A276569 Numbers such that ludic factor of n (A272565) does not divide n.

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A276437 Numbers n for which A020639(n) <> A272565(n).

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A302035 a(1) = 0, for n > 1, a(n) = A001511(A260739(n)); Number of instances of (the smallest) Ludic factor A272565(n) in n.

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A276447 Numbers n for which A272565(n) < A020639(n).

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A276448 Numbers n for which A272565(n) > A020639(n).

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A276570 a(n) = n modulo ludic factor of n: n mod A272565(n).

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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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