A050505 -id:A050505 - cp's OEIS frontend

A016969 a(n) = 6*n + 5.

5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 191, 197, 203, 209, 215, 221, 227, 233, 239, 245, 251, 257, 263, 269, 275, 281, 287, 293, 299, 305, 311, 317, 323, 329, 335

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(18).
Exponents e such that x^e + x - 1 is reducible.
First differences of A141631. - Paul Curtz, Sep 12 2008
a(n-1), n >= 1, appears as first column in the triangle A239127 related to the Collatz problem. - Wolfdieter Lang, Mar 14 2014
Odd unlucky numbers in A050505. - Fred Daniel Kline, Feb 25 2017
Intersection of A005408 and A016789. - Bruno Berselli, Apr 26 2018
Numbers that are not divisible by their digital root in base 4. - Amiram Eldar, Nov 24 2022

Muniru A Asiru, Table of n, a(n) for n = 0..3000
Mark W. Coffey, Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers, arXiv:1601.01673 [math.NT], 2016.
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 949.
D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., Vol. 36, No. 3 (1935), pp. 637-649.
Amelia Carolina Sparavigna, The Pentagonal Numbers and their Link to an Integer Sequence which contains the Primes of Form 6n-1, Politecnico di Torino (Italy, 2021).
Amelia Carolina Sparavigna, Binary operations inspired by generalized entropies applied to figurate numbers, Politecnico di Torino (Italy, 2021).
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Leo Tavares, Illustration: Twin Triangular Frames.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A111863, A007310, A008588, A016921, A016933, A016945, A016957, A049452.
Cf. A050505 (unlucky numbers).
Cf. A000217.

List([0..60],n->6*n+5); # Muniru A Asiru, Nov 24 2018

[ 6*n+5: n in [0..55] ]; // Klaus Brockhaus, Jan 04 2009

6Range[0, 59] + 5 (* or *) NestList[6 + # &, 5, 60] (* Harvey P. Dale, Mar 09 2013 *)

a(n)=6*n+5 \\ Charles R Greathouse IV, Jul 10 2016

(1 to 60).map(6 *  - 1).mkString(", ") // _Alonso del Arte, Nov 23 2018

a(n) = A003415(A003415(A125200(n+1)))/2. - Reinhard Zumkeller, Nov 24 2006
A008615(a(n)) = n+1. - Reinhard Zumkeller, Feb 27 2008
a(n) = A007310(2*n+1); complement of A016921 with respect to A007310. - Reinhard Zumkeller, Oct 02 2008
From Klaus Brockhaus, Jan 04 2009: (Start)
G.f.: (5+x)/(1-x)^2.
a(0) = 5; for n > 0, a(n) = a(n-1)+6. (End)
a(n) = A016921(n)+4 = A016933(n)+3 = A016945(n)+2 = A016957(n)+1. - Klaus Brockhaus, Jan 04 2009
a(n) = floor((12n-1)/2) with offset 1..a(1)=5. - Gary Detlefs, Mar 07 2010
a(n) = 4*(3*n+1) - a(n-1) (with a(0) = 5). - Vincenzo Librandi, Nov 20 2010
a(n) = floor(1/(1/sin(1/n) - n)). - Clark Kimberling, Feb 19 2010
a(n) = 3*Sum_{k = 0..n} binomial(6*n+5, 6*k+2)*Bernoulli(6*k+2). - Michel Marcus, Jan 11 2016
a(n) = A049452(n+1) / (n+1). - Torlach Rush, Nov 23 2018
a(n) = 2*A000217(n+2) - 1 - 2*A000217(n-1). See Twin Triangular Frames illustration. - Leo Tavares, Aug 25 2021
Sum_{n>=0} (-1)^n/a(n) = Pi/6 - sqrt(3)*arccoth(sqrt(3))/3. - Amiram Eldar, Dec 10 2021
E.g.f.: exp(x)*(5 + 6*x). - Stefano Spezia, Feb 14 2025

More terms from Klaus Brockhaus, Jan 04 2009

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))

A145649 Characteristic function of the lucky numbers.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1

Offset: 1

Reinhard Zumkeller, Oct 15 2008

nonn

Is there an efficient formula for this sequence? To wit, is there an algorithm for determining whether n is a lucky or unlucky number which is substantially faster than determining the lucky numbers up to n? - Charles R Greathouse IV, Nov 24 2021

Antti Karttunen, Table of n, a(n) for n = 1..100005
Eric Weisstein's World of Mathematics, Lucky Numbers
Index entries for characteristic functions
Index entries for sequences generated by sieves

Cf. A000959 (lucky numbers), A050505 (complement: unlucky numbers).

A260438 Row index to A255545: If n is k-th Lucky number then a(n) = k, otherwise a(n) = number of the stage where n is removed in Lucky sieve.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 1, 6, 1, 2, 1, 3, 1, 7, 1, 2, 1, 8, 1, 4, 1, 2, 1, 9, 1, 10, 1, 2, 1, 11, 1, 3, 1, 2, 1, 12, 1, 5, 1, 2, 1, 13, 1, 14, 1, 2, 1, 6, 1, 4, 1, 2, 1, 3, 1, 15, 1, 2, 1, 16, 1, 17, 1, 2, 1, 18, 1, 19, 1, 2, 1, 20, 1, 3, 1, 2, 1, 7, 1, 21, 1, 2, 1, 4, 1, 22, 1, 2, 1, 5, 1, 23, 1, 2, 1, 3, 1, 24, 1, 2, 1, 8, 1, 25, 1, 2, 1, 26, 1, 6, 1, 2, 1

Offset: 1

Antti Karttunen, Jul 29 2015

nonn

For n >= 2 this works also as a row index to array A255551 (which does not contain 1) and when restricted to unlucky numbers, A050505, also as a row index to array A255543.

Antti Karttunen, Table of n, a(n) for n = 1..10003

Cf. A000959, A050505, A109497, A145649, A219178, A255543, A255545, A255551.
Cf. also A260429, A260439 (corresponding column indices).
Cf. A055396, A260738 for row indices to other arrays similar to A255545.

(define (A260438 n) (cond ((not (zero? (A145649 n))) (A109497 n)) ((even? n) 1) (else (let searchrow ((row 2)) (let searchcol ((col 1)) (cond ((>= (A255543bi row col) n) (if (= (A255543bi row col) n) row (searchrow (+ 1 row)))) (else (searchcol (+ 1 col))))))))) ;; Code for A255543bi given in A255543.

Other identities. For all n >= 1:
a(A000959(n)) = n.
a(A219178(n)) = n.
a(2n) = 1. [All even numbers are removed at the stage one of the sieve.]
a(A016969(n)) = 2.
a(A258016(n)) = 3.
a(A260440(n)) = 4.
A255545(a(n), A260429(n)) = n.
For all n >= 2, A255551(a(n), A260439(n)) = n.

A260439 Column index to A255551: a(1) = 0; for n > 1: if n is Lucky number then a(n) = 1, otherwise for a(2k) = k, and for odd unlucky numbers, a(n) = 1 + the position at the stage where n is removed in the Lucky sieve.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 1, 4, 1, 5, 3, 6, 1, 7, 1, 8, 4, 9, 2, 10, 1, 11, 5, 12, 1, 13, 2, 14, 6, 15, 1, 16, 1, 17, 7, 18, 1, 19, 3, 20, 8, 21, 1, 22, 2, 23, 9, 24, 1, 25, 1, 26, 10, 27, 2, 28, 3, 29, 11, 30, 4, 31, 1, 32, 12, 33, 1, 34, 1, 35, 13, 36, 1, 37, 1, 38, 14, 39, 1, 40, 5, 41, 15, 42, 2, 43, 1, 44, 16, 45, 4, 46, 1, 47, 17, 48, 3, 49, 1, 50, 18, 51, 6, 52, 1

Offset: 1

Antti Karttunen, Jul 29 2015

nonn

a(1) = 0, because 1 is outside of A255551 array proper.

Antti Karttunen, Table of n, a(n) for n = 1..10003

Cf. A000959, A050505, A145649, A255543, A255551.
Cf. also A260438 (corresponding row index).
Cf. A078898, A246277, A260429, A260437, A260739 for column indices to other arrays similar to A255551.

(define (A260439 n) (cond ((= 1 n) 0) ((not (zero? (A145649 n))) 1) ((even? n) (/ n 2)) (else (let searchrow ((row 2)) (let searchcol ((col 1)) (cond ((>= (A255543bi row col) n) (if (= (A255543bi row col) n) (+ 1 col) (searchrow (+ 1 row)))) (else (searchcol (+ 1 col))))))))) ;; Code for A255543bi given in A255543.

Other identities. For all n >= 1:
a(2n) = n.
Also, for all n >= 2:
A255551(A260438(n), a(n)) = n.
a(A219178(n)) = 2.

A257801 Permutation of natural numbers: a(1)=1; a(oddprime(n)) = lucky(1+a(n)), a(not_an_oddprime(n)) = unlucky(a(n-1)).

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 9, 6, 11, 8, 13, 14, 25, 10, 17, 12, 15, 19, 33, 20, 35, 16, 21, 24, 18, 22, 27, 45, 43, 28, 31, 47, 23, 29, 34, 26, 51, 30, 38, 59, 63, 57, 115, 39, 42, 61, 37, 32, 40, 46, 36, 66, 73, 41, 52, 78, 83, 76, 49, 146, 67, 53, 56, 81, 50, 44, 79, 54, 60, 48, 163, 86, 87, 95, 55, 68, 101, 107, 171, 98, 64

Offset: 1

Antti Karttunen, May 09 2015

nonn

Here lucky(n) = n-th lucky number = A000959(n), unlucky(n) = n-th unlucky number = A050505(n), oddprime(n) = n-th odd prime = A065091(n), not_an_oddprime(n) = n-th natural number which is not an odd prime = A065090(n).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A257802.
Cf. A000959, A000720, A010051, A050505, A062298, A065090, A065091.
Related or similar permutations: A257726, A257727, A257730, A257731.

a(1) = 1; a(2) = 2; if A010051(n) = 1 [i.e., when n is an (odd) prime] then a(n) = A000959(1+a(A000720(n)-1)), otherwise a(n) = A050505(a(A062298(n))).
As a composition of other permutations:
a(n) = A257726(A257727(n)).
a(n) = A257731(A257730(n)).

A260429 Column index to A255545: if n is Lucky number, then a(n) = 1, otherwise a(n) = 1 + the position at the stage where n is removed in the Lucky sieve.

Original entry on oeis.org

1, 2, 1, 3, 2, 4, 1, 5, 1, 6, 3, 7, 1, 8, 1, 9, 4, 10, 2, 11, 1, 12, 5, 13, 1, 14, 2, 15, 6, 16, 1, 17, 1, 18, 7, 19, 1, 20, 3, 21, 8, 22, 1, 23, 2, 24, 9, 25, 1, 26, 1, 27, 10, 28, 2, 29, 3, 30, 11, 31, 4, 32, 1, 33, 12, 34, 1, 35, 1, 36, 13, 37, 1, 38, 1, 39, 14, 40, 1, 41, 5, 42, 15, 43, 2, 44, 1, 45, 16, 46, 4, 47, 1, 48, 17, 49, 3, 50, 1, 51, 18, 52, 6, 53, 1

Offset: 1

Antti Karttunen, Jul 29 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10003

One more than A260437.
Cf. A000959, A050505, A145649, A219178, A255543, A255545.
Cf. also A260438 (corresponding row index).
Cf. A078898, A246277, A260439, A260739 for column indices to other arrays similar to A255545.

(define (A260429 n) (cond ((not (zero? (A145649 n))) 1) ((even? n) (+ 1 (/ n 2))) (else (let searchrow ((row 2)) (let searchcol ((col 1)) (cond ((>= (A255543bi row col) n) (if (= (A255543bi row col) n) (+ 1 col) (searchrow (+ 1 row)))) (else (searchcol (+ 1 col))))))))) ;; Code for A255543bi given in A255543.

Other identities. For all n >= 1:
a(n) = 1 + A260437(n).
Iff A145649(n) = 1, then a(n) = 1.
a(2n) = n+1. [Even numbers are removed at the stage one of the sieve, after 1 which is also removed in the beginning.]
a(A219178(n)) = 2.
A255545(A260438(n), a(n)) = n.

A264940 Lucky factor of n.

Original entry on oeis.org

0, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 7, 2, 0, 2, 3, 2, 0, 2, 9, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 7, 2, 3, 2, 0, 2, 13, 2, 3, 2, 0, 2, 0, 2, 3, 2, 15, 2, 9, 2, 3, 2, 7, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 7, 2, 3, 2, 21, 2

Offset: 1

Max Barrentine, Dec 09 2015

nonn

This sequence is analogous to the smallest prime factor of n (A020639). If n is lucky, a(n)=0; if n is unlucky, a(n) is the number that rejects n from the lucky number sieve. This is 2 for even numbers, and a lucky number >= 3 for odd unlucky numbers.

Max Barrentine, Table of n, a(n) for n = 1..10000

Cf. A000959, A050505, A145649, A219178, A255543, A255545, A260438, A265859.

Cf. A020639, A271419 (somewhat analogous sequences).

(define (A264940 n) (if (= 1 (A145649 n)) 0 (if (even? n) 2 (A000959 (A260438 n))))) ;; Antti Karttunen, Sep 11 2016

From Antti Karttunen, Sep 11 2016: (Start)
If A145649(n) = 1 [when n is lucky], a(n) = 0, else if n is even, a(n) = 2, otherwise a(n) = A000959(A265859(n)) = A000959(A260438(n)).
For n >= 2, a(A219178(n)) = A000959(n).
(End)

Formula corrected and comment clarified by Antti Karttunen, Sep 11 2016

A257802 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = oddprime(a(n-1)), a(unlucky(n)) = not_an_oddprime(1+a(n)).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 10, 7, 14, 9, 16, 11, 12, 17, 22, 15, 25, 18, 20, 23, 26, 33, 24, 13, 36, 27, 30, 34, 38, 31, 48, 19, 35, 21, 51, 47, 39, 44, 49, 54, 45, 29, 66, 28, 50, 32, 70, 59, 65, 37, 55, 62, 68, 75, 63, 42, 90, 40, 69, 46, 94, 41, 81, 88, 52, 61, 76, 83, 85, 92, 100, 53, 86, 101, 58, 120, 56, 67, 93, 64

Offset: 1

Antti Karttunen, May 09 2015

nonn

Here lucky(n) = n-th lucky number = A000959(n), unlucky(n) = n-th unlucky number = A050505(n), oddprime(n) = n-th odd prime = A065091(n), not_an_oddprime(n) = n-th natural number which is not an odd prime = A065090(n).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A257801.
Cf. A000959, A050505, A065090, A065091, A109497, A145649.
Related or similar permutations: A257725, A257728, A257729, A257732.

a(1) = 1; if A145649(n) = 1 [i.e., when n is lucky] then a(n) = A065091(a(A109497(n)-1)), otherwise a(n) = A065090(1+a(n-A109497(n))).
As a composition of other permutations:
a(n) = A257728(A257725(n)).
a(n) = A257729(A257732(n)).

A260437 Column index to A255543: if n is Lucky number then a(n) = 0, otherwise a(n) = the position at the stage where n is removed in the Lucky sieve.

Original entry on oeis.org

0, 1, 0, 2, 1, 3, 0, 4, 0, 5, 2, 6, 0, 7, 0, 8, 3, 9, 1, 10, 0, 11, 4, 12, 0, 13, 1, 14, 5, 15, 0, 16, 0, 17, 6, 18, 0, 19, 2, 20, 7, 21, 0, 22, 1, 23, 8, 24, 0, 25, 0, 26, 9, 27, 1, 28, 2, 29, 10, 30, 3, 31, 0, 32, 11, 33, 0, 34, 0, 35, 12, 36, 0, 37, 0, 38, 13, 39, 0, 40, 4, 41, 14, 42, 1, 43, 0, 44, 15, 45, 3, 46, 0, 47, 16, 48, 2, 49, 0, 50, 17, 51, 5, 52

Offset: 1

Antti Karttunen, Aug 02 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10003

One less than A260429.
Cf. A000959, A050505, A145649, A219178, A255543.
Cf. also A260438 (corresponding row index).

(define (A260437 n) (cond ((not (zero? (A145649 n))) 0) ((even? n) (/ n 2)) (else (let searchrow ((row 2)) (let searchcol ((col 1)) (cond ((>= (A255543bi row col) n) (if (= (A255543bi row col) n) col (searchrow (+ 1 row)))) (else (searchcol (+ 1 col))))))))) ;; Code for A255543bi given in A255543.

Other identities. For all n >= 1:
a(n) = A260429(n) - 1.
Iff A145649(n) = 1, then a(n) = 0.
a(2n) = n.
a(A219178(n)) = 1.
A255543(A260438(A050505(n)), a(A050505(n))) = A050505(n).

cp's OEIS Frontend

A016969 a(n) = 6*n + 5.

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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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A145649 Characteristic function of the lucky numbers.

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A260438 Row index to A255545: If n is k-th Lucky number then a(n) = k, otherwise a(n) = number of the stage where n is removed in Lucky sieve.

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A260439 Column index to A255551: a(1) = 0; for n > 1: if n is Lucky number then a(n) = 1, otherwise for a(2k) = k, and for odd unlucky numbers, a(n) = 1 + the position at the stage where n is removed in the Lucky sieve.

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A257801 Permutation of natural numbers: a(1)=1; a(oddprime(n)) = lucky(1+a(n)), a(not_an_oddprime(n)) = unlucky(a(n-1)).

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A260429 Column index to A255545: if n is Lucky number, then a(n) = 1, otherwise a(n) = 1 + the position at the stage where n is removed in the Lucky sieve.

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A264940 Lucky factor of n.

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