A219178 -id:A219178 - cp's OEIS frontend

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

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).

A258207 Square array: row n gives the numbers remaining after the stage n of Lucky sieve.

Original entry on oeis.org

1, 3, 1, 5, 3, 1, 7, 7, 3, 1, 9, 9, 7, 3, 1, 11, 13, 9, 7, 3, 1, 13, 15, 13, 9, 7, 3, 1, 15, 19, 15, 13, 9, 7, 3, 1, 17, 21, 21, 15, 13, 9, 7, 3, 1, 19, 25, 25, 21, 15, 13, 9, 7, 3, 1, 21, 27, 27, 25, 21, 15, 13, 9, 7, 3, 1, 23, 31, 31, 31, 25, 21, 15, 13, 9, 7, 3, 1, 25, 33, 33, 33, 31, 25, 21, 15, 13, 9, 7, 3, 1, 27, 37, 37, 37, 33, 31, 25, 21, 15, 13, 9, 7, 3, 1, 29, 39, 43, 43, 37, 33, 31, 25, 21, 15, 13, 9, 7, 3, 1

Offset: 1

Antti Karttunen, Jul 27 2015

This square 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), etc.
Lucky sieve starts with natural numbers: 1, 2, 3, 4, 5, 6, 7, ... from which at first stage the even numbers are removed, and on each subsequent stage n (n > 1) one sets k =  (these k will form the Lucky numbers) and removes every k-th term (from column positions k, 2k, 3k, etc.) of the preceding row to produce the next row of this array.
On each row n, the first term that differs from the n-th Lucky number (A000959(n)) occurs at the column position A000959(n+1) and that number is A219178(n) when n > 1.

			The top left corner of the array:
1, 3, 5, 7,  9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39
1, 3, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 39, 43, 45, 49, 51, 55, 57
1, 3, 7, 9, 13, 15, 21, 25, 27, 31, 33, 37, 43, 45, 49, 51, 55, 57, 63, 67
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 45, 49, 51, 55, 63, 67, 69, 73
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 55, 63, 67, 69, 73, 75
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79
...
To get row 2 from row 1, we use the second term of the first row, which is 3, to remove every third term from row 1: 5, 11, 17, ... which leaves 1, 3, 7, 9, 13, ...
To get row 3 from row 2, we use the third term of row 2, which is 7, to remove every seventh term from row 2: 19, 39, ... which then results in the third row.

Cf. A000959 (Lucky numbers), which occur at the main and also any subdiagonal of this array. Also the rows converge towards A000959.
Row 1: A005408. Row 2: A047241. Row 3: A258011.
Transpose: A258208.
Cf. also A219178, A255543, A260717.

(define (A258207 n) (A258207bi (A002260 n) (A004736 n)))
(define (A258207bi row col) ((rowfun_n_for_A000959sieve row) col))
;; Uses definec-macro which can memoize also function-closures:
(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)))))))
(define (A005408shifted n) (- (* 2 n) 1))

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

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

Original entry on oeis.org

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).

cp's OEIS Frontend

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

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A258207 Square array: row n gives the numbers remaining after the stage n of Lucky sieve.

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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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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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