A255543 -id:A255543 - cp's OEIS frontend

A255550 Main diagonal of array A255551.

2, 5, 39, 91, 199, 315, 567, 829, 1227, 1513, 1953, 2569, 3277, 3769, 5119, 5925, 6607, 7539, 8319, 9375, 11007, 12511, 14103, 15801, 17593, 19165, 22213, 23617, 25467, 26967, 29347, 32733, 35809, 38085, 40953, 42915, 49093, 51787, 54055, 57459, 60409, 64057, 68433, 71637, 76299, 79719, 82545, 86133, 94921, 98037, 102745

Offset: 1

Antti Karttunen, Feb 26 2015

nonn

Equally, 2 followed by the first subdiagonal of A255543.

Cf. A255543, A255551, A255549.

Cf. also A083141, A255410, A255553.

a(n) = A255551(n,n).
a(1) = 2; for n > 1: a(n) = A255543(n,n-1).
Other identities.
For all n >= 1, a(n) = A255553(A083141(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

A265859 Unluckiness of n.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 3, 1, 0, 1, 2, 1, 0, 1, 4, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 3, 1, 2, 1, 0, 1, 5, 1, 2, 1, 0, 1, 0, 1, 2, 1, 6, 1, 4, 1, 2, 1, 3, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 3, 1, 2, 1, 7, 1, 0

Offset: 1

Max Barrentine, Dec 17 2015

nonn

If n is lucky, a(n)=0; if n is unlucky, a(n) is the step at which n is rejected from the lucky number sieve.

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

Cf. A000959, A050505, A145649, A260437, A260438.
Cf. also arrays A255543, A255545.
Cf. A271420 (analogous sequence).

(define (A265859 n) (* (- 1 (A145649 n)) (A260438 n))) ;; Antti Karttunen, Sep 11 2016

a(n) = (1-A145649(n)) * A260438(n). - Antti Karttunen, Sep 11 2016, corrected Nov 15 2019

A255544 Unlucky array, transposed.

Original entry on oeis.org

2, 5, 4, 19, 11, 6, 27, 39, 17, 8, 45, 57, 61, 23, 10, 55, 97, 91, 81, 29, 12, 85, 117, 147, 121, 103, 35, 14, 109, 177, 181, 199, 153, 123, 41, 16, 139, 225, 277, 243, 253, 183, 145, 47, 18, 157, 295, 345, 369, 315, 301, 217, 165, 53, 20, 175, 325, 447, 465, 471, 379, 351, 247, 187, 59, 22, 213, 381, 493, 603, 589, 567, 441, 403, 279, 207, 65, 24

Offset: 1

Antti Karttunen, Feb 25 2015

See comments at A255543.

Antti Karttunen, Table of n, a(n) for n = 1..5886; the first 108 antidiagonals of the array, flattened

Transpose: A255543.

Permutation of A050505.

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, 1, n}] // Flatten (* Jean-François Alcover, Mar 15 2016 *)

(define (A255544 n) (A255543bi (A004736 n) (A002260 n))) ;; Other code as in A255543.

A255547 Lucky / Unlucky array, transposed.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 25 2015

See comments at A255545.

Transpose: A255545.
Inverse: A255548.
Topmost row: A000959, under which follows the rows of array A255544.

(define (A255547 n) (A255545bi (A004736 n) (A002260 n))) ;; Other code as in A255545 and A255543.

A258016 Unlucky numbers removed at the stage three of Lucky sieve.

Original entry on oeis.org

19, 39, 61, 81, 103, 123, 145, 165, 187, 207, 229, 249, 271, 291, 313, 333, 355, 375, 397, 417, 439, 459, 481, 501, 523, 543, 565, 585, 607, 627, 649, 669, 691, 711, 733, 753, 775, 795, 817, 837, 859, 879, 901, 921, 943, 963, 985, 1005, 1027, 1047, 1069, 1089, 1111, 1131, 1153, 1173, 1195, 1215, 1237, 1257, 1279, 1299, 1321, 1341, 1363, 1383, 1405

Offset: 1

Antti Karttunen, Jul 27 2015

Numbers congruent to 19 or 39 modulo 42. - Jianing Song, Apr 27 2022

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1)

Row 3 of A255543. Every seventh term of A047241.

Cf. also A258011.

a(n) = A047241(7*n).
a(n) = A260436(A255413(1+n)).
From Jianing Song, Apr 27 2022: (Start)
a(n) = a(n-2) + 42.
a(n) = a(n-1) + a(n-2) - a(n-3).
G.f.: (19*x+20*x^2+3*x^3)/(1-x-x^2+x^3).
E.g.f.: 3 + (21*x-3)*cosh(x) + (21*x-2)*sinh(x). (End)

A260440 Unlucky numbers removed at the stage four of Lucky sieve.

Original entry on oeis.org

27, 57, 91, 121, 153, 183, 217, 247, 279, 309, 343, 373, 405, 435, 469, 499, 531, 561, 595, 625, 657, 687, 721, 751, 783, 813, 847, 877, 909, 939, 973, 1003, 1035, 1065, 1099, 1129, 1161, 1191, 1225, 1255, 1287, 1317, 1351, 1381, 1413, 1443, 1477, 1507, 1539, 1569, 1603, 1633, 1665, 1695, 1729, 1759, 1791, 1821, 1855, 1885, 1917, 1947, 1981, 2011

Offset: 1

Antti Karttunen, Jul 27 2015

Numbers congruent to {27, 57, 91, 121} modulo 126. - Jianing Song, Apr 27 2022

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1)

Row 4 of A255543. Every ninth term of A258011.

a(n) = A258011(9*n).
a(n) = A260436(A255414(1+n)).
From Jianing Song, Apr 27 2022: (Start)
a(n) = a(n-4) + 126.
a(n) = a(n-1) + a(n-4) - a(n-5).
G.f.: (27*x+30*x^2+34*x^3+30*x^4+5*x^5)/(1-x-x^4+x^5).
E.g.f: 1/2*(10 + cos(x) - sin(x) + (63*x-11)*cosh(x) + (63*x-8)*sinh(x)). (End)

cp's OEIS Frontend

A255550 Main diagonal of array A255551.

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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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A265859 Unluckiness of n.

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A255544 Unlucky array, transposed.

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A255547 Lucky / Unlucky array, transposed.

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A258016 Unlucky numbers removed at the stage three of Lucky sieve.

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A260440 Unlucky numbers removed at the stage four of Lucky sieve.

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