A255551 -id:A255551 - cp's OEIS frontend

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

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))

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.

A255554 Permutation of natural numbers: a(n) = A083221(A255552(n)).

Original entry on oeis.org

1, 2, 3, 4, 9, 6, 5, 8, 7, 10, 15, 12, 11, 14, 13, 16, 21, 18, 25, 20, 17, 22, 27, 24, 19, 26, 49, 28, 33, 30, 23, 32, 29, 34, 39, 36, 31, 38, 35, 40, 45, 42, 37, 44, 121, 46, 51, 48, 41, 50, 43, 52, 57, 54, 169, 56, 77, 58, 63, 60, 55, 62, 47, 64, 69, 66, 53, 68, 59, 70, 75, 72, 61, 74, 67, 76, 81, 78, 71, 80, 65, 82, 87, 84, 289, 86, 73, 88, 93, 90, 91, 92, 79

Offset: 1

Antti Karttunen, Feb 26 2015

nonn

a(n) tells which number in array A083221, constructed from the sieve of Eratosthenes is at the same position where n is in array A255551 constructed from Lucky sieve. As both arrays have A005843 (even numbers) as their topmost row, this permutation fixes all of them.

Index entries for sequences that are permutations of the natural numbers

Inverse: A255553.
Similar permutations: A255407, A255408, A249817, A249818.
Cf. A000040, A000959, A001248, A219178, A083141, A083221, A255550, A255552.

(define (A255554 n) (A083221 (A255552 n)))

a(n) = A083221(A255552(n)).
Other identities:
a(2n) = 2n. [Fixes even numbers.]
For all n >= 1, a(A255550(n)) = A083141(n).
For all n >= 2, a(A000959(n)) = A000040(n).
For all n >= 2, a(A219178(n)) = A001248(n).

A269370 a(1) = 1, after which, for odd n: a(n) = A260439(n)-th number k for which A260438(k) = A260438(n)-1, and for even n: a(n) = a(n/2).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 3, 1, 7, 4, 6, 2, 9, 3, 13, 1, 8, 7, 5, 4, 15, 6, 10, 2, 21, 9, 19, 3, 12, 13, 25, 1, 31, 8, 14, 7, 33, 5, 11, 4, 16, 15, 37, 6, 27, 10, 18, 2, 43, 21, 49, 9, 20, 19, 45, 3, 39, 12, 22, 13, 17, 25, 51, 1, 24, 31, 63, 8, 67, 14, 26, 7, 69, 33, 73, 5, 28, 11, 75, 4, 23, 16, 30, 15, 55, 37, 79, 6, 32, 27, 61, 10, 87, 18, 34, 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 A255551 (square array generated by Lucky sieve), in the same column where n itself is.

Antti Karttunen, Table of n, a(n) for n = 1..11017

Cf. A255551, A260438, A260439, A269369.

Cf. also A268674, A269380.

(define (A269370 n) (cond ((= 1 n) n) ((even? n) (A269370 (/ n 2))) (else (A255551bi (- (A260438 n) 1) (A260439 n))))) ;; Code for A255551bi given in A255551.

a(1) = 1; after which for even n, a(n) = a(n/2), and for odd n, a(n) = A255551(A260438(n)-1, A260439(n)).
Other identities. For all n >= 1:
a(A269369(n)) = n.

A260435 Permutation mapping from Lucky sieve to Ludic sieve: a(1) = 1, for n > 1: a(n) = A255127(A260438(n), A260439(n)).

Original entry on oeis.org

1, 2, 3, 4, 9, 6, 5, 8, 7, 10, 15, 12, 11, 14, 13, 16, 21, 18, 19, 20, 17, 22, 27, 24, 23, 26, 31, 28, 33, 30, 25, 32, 29, 34, 39, 36, 37, 38, 35, 40, 45, 42, 41, 44, 55, 46, 51, 48, 43, 50, 47, 52, 57, 54, 73, 56, 59, 58, 63, 60, 49, 62, 53, 64, 69, 66, 61, 68, 67, 70, 75, 72, 71, 74, 77, 76, 81, 78, 83, 80, 65, 82, 87, 84, 101, 86, 89, 88

Offset: 1

Antti Karttunen, Jul 30 2015

nonn

a(n) tells which number in array A255127 (constructed from Ludic sieve) is at the same position where n is in array A255551 (constructed from Lucky sieve). This permutation fixes all even numbers because both arrays have A005843 as their topmost row.

Inverse: A260436.
Cf. A000959, A003309, A255127, A260438, A260439.
Similar or related permutations: A255407, A255552, A255554, A249817, A249818, A260741 (a more recursed variant).

(define (A260435 n) (if (<= n 1) n (A255127bi (A260438 n) (A260439 n)))) ;; Code for A255127bi given in A255127.

Other identities. For all n >= 1:
a(A000959(n+1)) = A003309(n+2). [Maps Lucky numbers to odd Ludic numbers.]
a(2n) = 2n.
As a composition of related permutations:
a(n) = A255127(A255552(n)).
a(n) = A255407(A255554(n)).

A278511 Square array constructed from Flavius sieve, shifted version, read by descending antidiagonals.

Original entry on oeis.org

2, 4, 3, 6, 5, 7, 8, 11, 9, 13, 10, 17, 21, 15, 19, 12, 23, 33, 37, 25, 27, 14, 29, 45, 55, 51, 31, 39, 16, 35, 57, 75, 85, 73, 43, 49, 18, 41, 69, 97, 111, 121, 99, 61, 63, 20, 47, 81, 115, 145, 159, 151, 127, 67, 79, 22, 53, 93, 135, 171, 199, 211, 193, 163, 87, 91, 24, 59, 105, 157, 205, 243, 267, 271, 247, 187, 103, 109

Offset: 1

Antti Karttunen, Nov 23 2016

Note how in comparison to A278505, 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 by cross-referencing. (See also A255551.)

			The top left corner of the array:
   2,  4,   6,   8,  10,  12,  14,  16,  18,  20
   3,  5,  11,  17,  23,  29,  35,  41,  47,  53
   7,  9,  21,  33,  45,  57,  69,  81,  93, 105
  13, 15,  37,  55,  75,  97, 115, 135, 157, 175
  19, 25,  51,  85, 111, 145, 171, 205, 231, 265
  27, 31,  73, 121, 159, 199, 243, 283, 327, 367
  39, 43,  99, 151, 211, 267, 319, 379, 433, 487
  49, 61, 127, 193, 271, 343, 421, 483, 559, 631
  63, 67, 163, 247, 339, 427, 519, 607, 691, 793
  79, 87, 187, 303, 403, 523, 639, 739, 853, 963

Antti Karttunen, Table of n, a(n) for n = 1..10440 [the first 144 antidiagonals of the array; offset adapted by _Georg Fischer_, Jul 15 2020]
Index entries for sequences generated by sieves
Index entries for sequences related to the Josephus Problem

Inverse: A278512.
Cf. A000960 (column 1, but with its initial 1 replaced by 2), A278505, A278507.
Cf. A278538 (row index of n), A278537 (column index of n).
Cf. A083221, A255127, A255551 (analogous arrays constructed from other sieves).

(define (A278511 n) (if (<= n 1) n (A278511bi (A002260 (- n 1)) (A004736 (- n 1)))))
(define (A278511bi row col) (cond ((= 1 row) (+ col col)) ((= 1 col) (A000960 row)) (else (A278507bi row (- col 1)))))
;; Code for A278507bi given in A278507.

A(1,col) = 2*col; For row > 1, A(row,1) = A000960(row) if col = 1, otherwise, A(row,col) = A278507(row,col-1).

For all n > 1, A(A278538(n), A278537(n)) = n.

cp's OEIS Frontend

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

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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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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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A255554 Permutation of natural numbers: a(n) = A083221(A255552(n)).

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A269370 a(1) = 1, after which, for odd n: a(n) = A260439(n)-th number k for which A260438(k) = A260438(n)-1, and for even n: a(n) = a(n/2).

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A260435 Permutation mapping from Lucky sieve to Ludic sieve: a(1) = 1, for n > 1: a(n) = A255127(A260438(n), A260439(n)).

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A278511 Square array constructed from Flavius sieve, shifted version, read by descending antidiagonals.

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