A255543 -id:A255543 - cp's OEIS frontend

A255549 Main diagonal of Unlucky array: a(n) = A255543(n,n).

2, 11, 61, 121, 253, 379, 663, 945, 1377, 1687, 2143, 2803, 3543, 4053, 5485, 6315, 7023, 7987, 8781, 9871, 11559, 13105, 14743, 16485, 18337, 19959, 23073, 24511, 26371, 27895, 30355, 33789, 36925, 39273, 42177, 44167, 50455, 53221, 55447, 58935, 61921, 65625, 70065, 73321, 78031, 81513, 84361, 87963, 96937, 100075

Offset: 1

Antti Karttunen, Feb 26 2015

nonn

a(n) = the n-th number in the infinite subset of unlucky numbers removed at the stage n in the lucky number sieve.

Cf. A219178, A255543, A255550.

rows = cols = 50; L = 2 Range[0, 10^5] + 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, n]], {n, 1, Min[rows, cols]}] (* Jean-François Alcover, Mar 15 2016 *)

(define (A255549 n) (A255543bi n n)) ;; Code for A255543bi given in A255543.

a(n) = A255543(n,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).

A000959 Lucky numbers.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, 105, 111, 115, 127, 129, 133, 135, 141, 151, 159, 163, 169, 171, 189, 193, 195, 201, 205, 211, 219, 223, 231, 235, 237, 241, 259, 261, 267, 273, 283, 285, 289, 297, 303

Offset: 1

N. J. A. Sloane; entry updated Mar 07 2008

An interesting general discussion of the phenomenon of 'random primes' (generalizing the lucky numbers) occurs in Hawkins (1958). Heyde (1978) proves that Hawkins' random primes do not only almost always satisfy the Prime Number Theorem but also the Riemann Hypothesis. - Alf van der Poorten, Jun 27 2002
Bui and Keating establish an asymptotic formula for the number of k-difference twin primes, and more generally to all l-tuples, of Hawkins primes, a probabilistic model of the Eratosthenes sieve. The formula for k = 1 was obtained by Wunderlich [Acta Arith. 26 (1974), 59 - 81]. - Jonathan Vos Post, Mar 24 2009. (This is quoted from the abstract of the Bui-Keating (2006) article, Joerg Arndt, Jan 04 2014)
It appears that a 1's line is formed, as in the Gilbreath's conjecture, if we use 2 (or 4), 3, 5 (differ of 7), 9, 13, 15, 21, 25, ... instead of A000959 1, 3, 7, 9, 13, 15, 21, 25, ... - Eric Desbiaux, Mar 25 2010
The Mersenne primes 2^p - 1 (= A000668, p in A000043) are in this sequence for p = 2, 3, 5, 7, 13, 17, and 19, but not for the following exponents p = 31, 61, and 89. - M. F. Hasler, May 06 2025

Martin Gardner, Gardner's Workout, Chapter 21 "Lucky Numbers and 2187" pp. 149-156 A. K. Peters MA 2002.
Richard K. Guy, Unsolved Problems in Number Theory, C3.
C. S. Ogilvy, Tomorrow's Math. 2nd ed., Oxford Univ. Press, 1972, p. 99.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. L. Stein and P. R. Stein, Tables of the Number of Binary Decompositions of All Even Numbers Less Than 200,000 into Prime Numbers and Lucky Numbers. Report LA-3106, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Sep 1964.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 116.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 114.

Hugo van der Sanden, Table of n, a(n) for n = 1..200000 (terms 1..10000 from T. D. Noe, terms 10001..30981 from R. J. Mathar)
José Juan Barco, Highly efficient C implementation with O(n log n) time and O(n) space (n bits) using hierarchical Fenwick trees and bitmaps to optimize memory access. Based on: J. Bille, I. Larsen, I. L. Gørtz et al., "Succinct Partial Sums and Fenwick Trees," ESA 2017.
José Juan Barco, Very fast O(n log n) sequence generating program in Haskell
H. M. Bui and J. P. Keating, On twin primes associated with the Hawkins random sieve, J. Number Theory 119(2) (2006), 284-296.
H. M. Bui and J. P. Keating, On twin primes associated with the Hawkins random sieve, arXiv:math/0607196 [math.NT], 2006-2010.
Vema Gardiner, R. Lazarus, N. Metropolis, and S. Ulam, On certain sequences of integers defined by sieves, Math. Mag., 29 (1956), 117-122. doi:10.2307/3029719; Zbl 0071.27002.
Martin Gardner, Lucky numbers and 2187, Math. Intellig., 19(2) (1997), 26-29.
David Hawkins, The random sieve, Math. Mag. 31 (1958), 1-3.
D. Hawkins and W. E. Briggs, The lucky number theorem, Math. Mag. 31 (1958), 81-84.
C. C. Heyde, A Log Log Improvement to the Riemann Hypothesis for the Hawkins Random Sieve, Ann. Probability, 6 (1978), 850-875.
Ivars Peterson and MathTrek, Martin Gardner's Lucky Numbers (archived on Archive.org).
Ivars Peterson, Martin Gardner's Lucky Numbers (archived on Wikiwix.com)
Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. This is Sieve #7. [Annotated and scanned copy]
Walter Schneider, Lucky Numbers, personal web site "Mathews: the Archive of Recreational Mathematics" (no more available: snapshot on web.archive.org as of May 2007), updated Dec 24, 2002.
Torsten Sillke, S. M. Ulam's Lucky Numbers
Hugo van der Sanden, Lucky numbers up to 1e8. [Broken link]
G. Villemin's Almanach of Numbers, Nombre Chanceux.
Eric Weisstein's World of Mathematics, Lucky number.
Wikipedia, Lucky number.
David W. Wilson, Fast space-efficient sequence generating program in C++.
Index entries for "core" sequences
Index entries for sequences related to lucky numbers
Index entries for sequences generated by sieves [From _Reinhard Zumkeller_, Oct 15 2008]

Main diagonal of A258207.
Column 1 of A255545. (cf. also arrays A255543, A255551).
Cf. A050505 (complement).
Cf. A145649 (characteristic function).
Cf. A137164-A137185, A039672, A045954, A249876.
Cf. A031883 (first differences), A254967 (iterated absolute differences), see also A054978.
Cf. A109497 (works as a left inverse function).
The Gilbreath transform is A054978 - see also A362460, A362461, A362462.
Cf. also A000040, A003309, A032600, A219178, A255553, A264940, A265859.

a000959 n = a000959_list !! (n-1)
a000959_list =  1 : sieve 2 [1,3..] where
   sieve k xs = z : sieve (k + 1) (lucky xs) where
      z = xs !! (k - 1 )
      lucky ws = us ++ lucky vs where
            (us, _:vs) = splitAt (z - 1) ws
-- Reinhard Zumkeller, Dec 05 2011

-- Also see links.
(C++) // See Wilson link, Nov 14 2012

## luckynumbers(n) returns all lucky numbers from 1 to n. ## Try n=10^5 just for fun. luckynumbers:=proc(n) local k, Lnext, Lprev; Lprev:=[$1..n]; for k from 1 do if k=1 or k=2 then Lnext:= map(w-> Lprev[w],remove(z -> z mod Lprev[2] = 0,[$1..nops(Lprev)])); if nops(Lnext)=nops(Lprev) then break fi; Lprev:=Lnext; else Lnext:= map(w-> Lprev[w],remove(z -> z mod Lprev[k] = 0,[$1..nops(Lprev)])); if nops(Lnext)=nops(Lprev) then break fi; Lprev:=Lnext; fi; od; return Lnext; end: # Walter Kehowski, Jun 05 2008; typo fixed by Robert Israel, Nov 19 2014
# Alternative
A000959List := proc(mx) local i, L, n, r;
L:= [seq(2*i+1, i=0..mx)]:
for n from 2 while n < nops(L) do
  r:= L[n];
  L:= subsop(seq(r*i=NULL, i=1..nops(L)/r), L);
od: L end:
A000959List(10^3); # Robert Israel, Nov 19 2014

luckies = 2*Range@200 - 1; f[n_] := Block[{k = luckies[[n]]}, luckies = Delete[luckies, Table[{k}, {k, k, Length@luckies, k}]]]; Do[f@n, {n, 2, 30}]; luckies (* Robert G. Wilson v, May 09 2006 *)
sieveMax = 10^6; luckies = Range[1, sieveMax, 2]; sieve[n_] := Module[{k = luckies[[n]]}, luckies = Delete[luckies, Table[{i}, {i, k, Length[luckies], k}]]]; n = 1; While[luckies[[n]] < Length[luckies], n++; sieve[n]]; luckies
L = Table[2*i + 1, {i, 0, 10^3}]; For[n = 2, n < Length[L], r = L[[n++]]; L = ReplacePart[L, Table[r*i -> Nothing, {i, 1, Length[L]/r}]]]; L (* Jean-François Alcover, Mar 15 2016, after Robert Israel *)

A000959_upto(nMax)={my(v=vectorsmall(nMax\2,k,2*k-1),i=1,q);while(v[i++]<=#v,v=vecextract(v,2^#v-1-(q=1<M. F. Hasler, Sep 22 2013, improved Jan 20 2020

def lucky(n):
    L = list(range(1, n + 1, 2))
    j = 1
    while j <= len(L) - 1 and L[j] <= len(L):
        del L[L[j]-1::L[j]]
        j += 1
    return L
# Robert FERREOL, Nov 19 2014, corrected by F. Chapoton, Mar 29 2020, performance improved by Ely Golden, Aug 18 2022

(define (A000959 n) ((rowfun_n_for_A000959sieve n) n)) ;; Code for rowfun_n_for_A000959sieve given in A255543.
;; Antti Karttunen, Feb 26 2015

Start with the natural numbers. Delete every 2nd number, leaving 1 3 5 7 ...; the 2nd number remaining is 3, so delete every 3rd number, leaving 1 3 7 9 13 15 ...; now delete every 7th number, leaving 1 3 7 9 13 ...; now delete every 9th number; etc.
a(n) = A254967(n-1, n-1). - Reinhard Zumkeller, Feb 11 2015
a(n) = A258207(n,n). [Where A258207 is a square array constructed from the numbers remaining after each step described above.] - Antti Karttunen, Aug 06 2015
A145649(a(n)) = 1; complement of A050505. - Reinhard Zumkeller, Oct 15 2008
Other identities from Antti Karttunen, Feb 26 2015: (Start)
For all n >= 1, A109497(a(n)) = n.
For all n >= 1, a(n) = A000040(n) + A032600(n).
For all n >= 2, a(n) = A255553(A000040(n)). (End)

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)

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).

A219178 a(n) = first unlucky number removed at the n-th stage of Lucky sieve.

Original entry on oeis.org

2, 5, 19, 27, 45, 55, 85, 109, 139, 157, 175, 213, 255, 265, 337, 363, 387, 411, 423, 457, 513, 547, 597, 637, 675, 715, 789, 807, 843, 871, 907, 987, 1033, 1083, 1113, 1125, 1267, 1297, 1315, 1371, 1407, 1465, 1515, 1555, 1609, 1651, 1671, 1707, 1851, 1873, 1927, 1969

Offset: 1

Phil Carmody, Nov 15 2012

First numbers removed by each lucky number in the lucky number sieve. - This is the original definition of the sequence, still valid from a(2) onward.

a(1) = 2, because at the first stage of Lucky sieve, all even numbers are removed, of which 2 is the first one. - Antti Karttunen, Feb 26 2015

			1 and 2 are a special case in the lucky number sieve, (1 is the lucky number, but every 2nd element is removed) so are ignored [in the original version of the sequence, which started from a(2). Now we have a(1) = 2. - _Antti Karttunen_, Feb 26 2015]. The 2nd lucky number, 3, removes { 5, 11, ... } from the list, so a(2) = 5. The 3rd lucky number, 7, removes { 19, 39, ... } from the list, so a(3)=19.

Antti Karttunen, Table of n, a(n) for n = 1..333
Wikipedia, Lucky Number

Column 1 of A255543, Column 2 of A255545 (And apart from the first term, also column 2 of A255551).

Cf. A000959, A001248, A254100, A255549, A255550, A255553.

rows = 52; cols = 1; L = 2 Range[0, 10^4] + 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, 1]], {n, 1, rows}] (* Jean-François Alcover, Mar 15 2016 *)

(define (A219178 n) (A255543bi n 1)) ;; Code for A255543bi given in A255543.

From Antti Karttunen, Feb 26 2015: (Start)
a(n) = A255543(n,1).
Other identities.
For all n >= 2, a(n) = A255553(A001248(n)).
(End)

Term a(1) = 2 prepended, without changing the rest of sequence. Name changed, with the original, more restrictive definition moved to the Comments section. - Antti Karttunen, Feb 26 2015

A255545 Lucky / Unlucky array: Each row starts with n-th lucky number, followed by all unlucky numbers removed at stage n of the sieve.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 11, 19, 9, 8, 17, 39, 27, 13, 10, 23, 61, 57, 45, 15, 12, 29, 81, 91, 97, 55, 21, 14, 35, 103, 121, 147, 117, 85, 25, 16, 41, 123, 153, 199, 181, 177, 109, 31, 18, 47, 145, 183, 253, 243, 277, 225, 139, 33, 20, 53, 165, 217, 301, 315, 369, 345, 295, 157, 37, 22, 59, 187, 247, 351, 379, 471, 465, 447, 325, 175, 43

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), ...

			The top left corner of the array:
   1,   2,   4,   6,   8,  10,  12,   14,   16,   18,   20,   22,   24,   26,   28
   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
...

Inverse: A255546.
Transpose: A255547.
Column 1: A000959. Other columns of array as in A255543, e.g. column 2: A219178.
Row 1: A004275 (starting from 1).
See A255551 for a slightly different variant.

(define (A255545 n) (A255545bi (A002260 n) (A004736 n)))
(define (A255545bi row col) (if (= 1 col) (A000959 row) (A255543bi row (- col 1))))
;; Other code as in A255543.

For col = 1, A(row,col) = A000959(row); otherwise, A(row,col) = A255543(row,col-1).

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.

A278507 Square array A(row,col) where row n lists the numbers removed in round n of Flavius sieve. 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, 5, 6, 11, 9, 8, 17, 21, 15, 10, 23, 33, 37, 25, 12, 29, 45, 55, 51, 31, 14, 35, 57, 75, 85, 73, 43, 16, 41, 69, 97, 111, 121, 99, 61, 18, 47, 81, 115, 145, 159, 151, 127, 67, 20, 53, 93, 135, 171, 199, 211, 193, 163, 87, 22, 59, 105, 157, 205, 243, 267, 271, 247, 187, 103, 24, 65, 117, 175, 231, 283, 319, 343, 339, 303, 229, 123

Offset: 1

Antti Karttunen, Nov 23 2016

			The top left corner of the array:
   2,   4,   6,   8,  10,  12,  14,  16,  18,   20
   5,  11,  17,  23,  29,  35,  41,  47,  53,   59
   9,  21,  33,  45,  57,  69,  81,  93, 105,  117
  15,  37,  55,  75,  97, 115, 135, 157, 175,  195
  25,  51,  85, 111, 145, 171, 205, 231, 265,  291
  31,  73, 121, 159, 199, 243, 283, 327, 367,  409
  43,  99, 151, 211, 267, 319, 379, 433, 487,  547
  61, 127, 193, 271, 343, 421, 483, 559, 631,  699
  67, 163, 247, 339, 427, 519, 607, 691, 793,  879
  87, 187, 303, 403, 523, 639, 739, 853, 963, 1081

Transpose: A278508.
This is array A278505 without its leftmost column, A000960.
Column 1: A100287 (apart from its initial 1).
Rows: A005843, A016969, A017629.
Cf. A278529 (column index of n), A278538 (row index of n).
Cf. A278492.
Cf. A255543 for analogous array for Lucky sieve.

(define (A278507 n) (A278507bi (A002260 n) (A004736 n)))
(define (A278507bi row col) (cond ((= 1 row) (* 2 col)) (else (A278492bi (- row 1) (+ -1 (* col (+ 1 row)))))))
;; Code for A278492bi given in A278492.

A(1,col) = 2*col; for row > 1, A(row,col) = A278492(row-1,(col*(row+1))-1). [Note that unlike this array, A278492 uses zero-based indexing for its rows and columns.]

cp's OEIS Frontend

A255549 Main diagonal of Unlucky array: a(n) = A255543(n,n).

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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.

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A000959 Lucky numbers.

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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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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), ...

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A219178 a(n) = first unlucky number removed at the n-th stage of Lucky sieve.

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A255545 Lucky / Unlucky array: Each row starts with n-th lucky number, followed by all unlucky numbers removed at stage n of the sieve.

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