A255127 -id:A255127 - cp's OEIS frontend

A255408 Permutation of natural numbers: a(n) = A083221(A255128(n)).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 25, 20, 21, 22, 19, 24, 23, 26, 27, 28, 29, 30, 49, 32, 33, 34, 35, 36, 31, 38, 39, 40, 37, 42, 41, 44, 45, 46, 43, 48, 55, 50, 51, 52, 47, 54, 121, 56, 57, 58, 77, 60, 53, 62, 63, 64, 65, 66, 59, 68, 69, 70, 61, 72, 169, 74, 75, 76, 67, 78, 85, 80, 81, 82, 71, 84, 91, 86, 87

Offset: 1

Antti Karttunen, Feb 22 2015

nonn

a(n) tells which number in array A083221 (the sieve of Eratosthenes) is at the same position where n is in Ludic array A255127. As both arrays have A005843 (even numbers) and A016945 as their two topmost rows, both sequences are among the fixed points of this permutation.

Equally: a(n) tells which number in array A083140 is at the same position where n is in the array A255129, as they are the transposes of above two arrays.

			A255127(3,2) = 19 and A083221(3,2) = 25, thus a(19) = 25.
A255127(8,1) = 23 and A083221(8,1) = 19, thus a(23) = 19.
A255127(9,1) = 25 and A083221(9,1) = 23, thus a(25) = 23.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A255407.
Cf. A003309, A008578, A083221, A255127, A255128.
Similar permutations: A249817.

(define (A255408 n) (A083221 (A255128 n)))

a(n) = A083221(A255128(n)).
Other identities. For all n >= 1:
a(2n) = 2n. [Fixes even numbers.]
a(3n) = 3n. [Fixes multiples of three.]
a(A003309(n)) = A008578(n). [Maps Ludic numbers to noncomposites.]

A269171 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A269379(a(A268674(2n+1))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 23, 20, 21, 22, 25, 24, 19, 26, 27, 28, 29, 30, 37, 32, 33, 34, 35, 36, 41, 46, 39, 40, 43, 42, 47, 44, 45, 50, 53, 48, 31, 38, 51, 52, 61, 54, 49, 56, 57, 58, 67, 60, 71, 74, 63, 64, 65, 66, 77, 68, 69, 70, 83, 72, 89, 82, 75, 92, 59, 78, 91, 80, 81, 86, 97, 84, 79, 94, 87, 88, 107, 90, 85, 100

Offset: 1

Antti Karttunen, Mar 03 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A269172.
Cf. A000035, A003309, A008578, A055396, A078898, A268674, A255127, A269379.
Related or similar permutations: A260741, A260742, A269355, A269357, A255421, A252754, A252756, A269385, A269387.
Cf. also A269393 (a(3n)/3) and A269395.
Differs from A255407 for the first time at n=38, where a(38) = 46, while A255407(38) = 38.

a(1) = 1, then after for even n, a(n) = 2*a(n/2), and for odd n, a(n) = A269379(a(A268674(n))).
a(1) = 1, for n > 1, a(n) = A255127(A055396(n), a(A078898(n))).
As a composition of other permutations:
a(n) = A269385(A252756(n)).
a(n) = A269387(A252754(n)).
Other identities. For all n >= 1:
A000035(a(n)) = A000035(n). [Preserves the parity of n.]
a(A008578(n)) = A003309(n). [Maps noncomposites to Ludic numbers.]

A260717 Square array: row n gives the numbers remaining before the stage n of Ludic sieve.

Original entry on oeis.org

2, 3, 3, 4, 5, 5, 5, 7, 7, 7, 6, 9, 11, 11, 11, 7, 11, 13, 13, 13, 13, 8, 13, 17, 17, 17, 17, 17, 9, 15, 19, 23, 23, 23, 23, 23, 10, 17, 23, 25, 25, 25, 25, 25, 25, 11, 19, 25, 29, 29, 29, 29, 29, 29, 29, 12, 21, 29, 31, 37, 37, 37, 37, 37, 37, 37, 13, 23, 31, 37, 41, 41, 41, 41, 41, 41, 41, 41, 14, 25, 35, 41, 43, 43, 43, 43, 43, 43, 43, 43, 43

Offset: 1

Antti Karttunen, Jul 30 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.

Ludic sieve starts with natural numbers larger than one: 2, 3, 4, 5, 6, 7, ... and in each subsequent stage one sets k = (which will be one of Ludic numbers) and removes both k and every k-th term after it, from column positions 1, 1+k, 1+2k, 1+3k, etc. of the preceding row to produce the next row of this array.

			The top left corner of the array:
   2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12,  13,  14,  15,  16,  17
   3,  5,  7,  9, 11, 13, 15, 17,  19,  21,  23,  25,  27,  29,  31,  33
   5,  7, 11, 13, 17, 19, 23, 25,  29,  31,  35,  37,  41,  43,  47,  49
   7, 11, 13, 17, 23, 25, 29, 31,  37,  41,  43,  47,  53,  55,  59,  61
  11, 13, 17, 23, 25, 29, 37, 41,  43,  47,  53,  55,  61,  67,  71,  73
  13, 17, 23, 25, 29, 37, 41, 43,  47,  53,  61,  67,  71,  73,  77,  83
  17, 23, 25, 29, 37, 41, 43, 47,  53,  61,  67,  71,  77,  83,  89,  91
  23, 25, 29, 37, 41, 43, 47, 53,  61,  67,  71,  77,  83,  89,  91,  97
  25, 29, 37, 41, 43, 47, 53, 61,  67,  71,  77,  83,  89,  91,  97, 107
  29, 37, 41, 43, 47, 53, 61, 67,  71,  77,  83,  89,  91,  97, 107, 115
  37, 41, 43, 47, 53, 61, 67, 71,  77,  83,  89,  91,  97, 107, 115, 119
  41, 43, 47, 53, 61, 67, 71, 77,  83,  89,  91,  97, 107, 115, 119, 121
  43, 47, 53, 61, 67, 71, 77, 83,  89,  91,  97, 107, 115, 119, 121, 127
  47, 53, 61, 67, 71, 77, 83, 89,  91,  97, 107, 115, 119, 121, 127, 131
  53, 61, 67, 71, 77, 83, 89, 91,  97, 107, 115, 119, 121, 127, 131, 143
  61, 67, 71, 77, 83, 89, 91, 97, 107, 115, 119, 121, 127, 131, 143, 149
  etc.

Transpose: A260718.
Column 1: A003309 (without the initial 1).
Row 1: A020725, Row 2: A144396, Row 3: A007310 (from its second term onward), Row 4: A260714, Row 5: A260715.
Cf. A255127 (gives the numbers removed at each stage).
Cf. also array A258207.

(define (A260717 n) (A260717bi (A002260 n) (A004736 n)))
(define (A260717bi row col) ((rowfun_n_for_A003309sieve row) col))
(define (add1 n) (1+ n))
;; Uses definec-macro which can memoize also function-closures:
(definec (rowfun_n_for_A003309sieve n) (if (= 1 n) add1 (let* ((prevrowfun (rowfun_n_for_A003309sieve (- n 1))) (everynth (prevrowfun 1))) (compose-funs prevrowfun (nonzero-pos 1 1 (lambda (i) (modulo (- i 1) everynth)))))))

A302036 Ludic powers: numbers k such that A302031(k) < 2; numbers k such that A260739(k) is a power of 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 29, 31, 32, 37, 41, 43, 45, 47, 49, 53, 55, 61, 64, 67, 71, 73, 77, 83, 85, 89, 91, 93, 97, 101, 107, 109, 115, 119, 121, 127, 128, 131, 143, 145, 149, 151, 157, 161, 167, 173, 175, 179, 181, 189, 191, 193, 197, 205, 209, 211, 221, 223, 227, 229, 233, 235, 239, 247, 253, 256, 257

Offset: 1

Antti Karttunen, Apr 02 2018

nonn

An analog of A000961 for factorization process based on the Ludic sieve (A255127).

Numbers k for which A302031(k) < 2, or equally, for which A302034(k) = 1, or equally, for which A209229(A260739(k)) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..3600
Index entries for sequences generated by sieves

Cf. A260739, A302031, A302034.
Cf. A000079, A003309, A254100 (subsequences).
Cf. also A000961, A302038, A302040.

for(n=1,257,if(A302031(n)<2,print1(n,","))); \\ See also code in A302031.

A255129 Transposed Ludic array.

Original entry on oeis.org

2, 3, 4, 5, 9, 6, 7, 19, 15, 8, 11, 31, 35, 21, 10, 13, 55, 59, 49, 27, 12, 17, 73, 103, 85, 65, 33, 14, 23, 101, 133, 151, 113, 79, 39, 16, 25, 145, 187, 197, 203, 137, 95, 45, 18, 29, 167, 271, 281, 263, 251, 163, 109, 51, 20, 37, 205, 311, 403, 367, 325, 299, 191, 125, 57, 22, 41, 253, 371, 457, 523, 461, 385, 343, 217, 139, 63, 24

Offset: 2

Antti Karttunen, Feb 22 2015

See the comments in A255127.

Antti Karttunen, Table of n, a(n) for n = 2..10441; the first 144 antidiagonals of array, flattened

Transpose: A255127.
Inverse: A255130. (When considered as a permutation of natural numbers with a(1) = 1).
Row 1: A003309 (without the initial 1).
Column 1: A005843 (even numbers).
Main diagonal: A255410.

(define (A255129 n) (if (<= n 1) n (A255129bi (A002260 (- n 1)) (A004736 (- n 1)))))
(define (A255129bi row col) ((rowfun_n_for_A255127 col) row))  ; Other code as in A255127.

A260436 Permutation mapping from Ludic sieve to Lucky sieve: a(1) = 1, for n > 1: a(n) = A255551(A260738(n), A260739(n)).

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 9, 8, 5, 10, 13, 12, 15, 14, 11, 16, 21, 18, 19, 20, 17, 22, 25, 24, 31, 26, 23, 28, 33, 30, 27, 32, 29, 34, 39, 36, 37, 38, 35, 40, 43, 42, 49, 44, 41, 46, 51, 48, 61, 50, 47, 52, 63, 54, 45, 56, 53, 58, 57, 60, 67, 62, 59, 64, 81, 66, 69, 68, 65, 70, 73, 72, 55, 74, 71, 76, 75, 78, 103, 80, 77, 82, 79, 84, 91, 86, 83, 88

Offset: 1

Antti Karttunen, Jul 30 2015

nonn

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

Inverse: A260435.
Cf. A000959, A003309, A255551, A260738, A260739.
Similar permutations: A255408, A255128, A255551, A255553, A249817, A249818, A260742 (a more recursed variant).

(define (A260436 n) (if (<= n 1) n (A255551bi (A260738 n) (A260739 n)))) ;; Code for A255551bi given in A255551.

Other identities. For all n >= 1:
a(A003309(n+2)) = A000959(n+1). [Maps odd Ludic numbers to Lucky numbers.]
a(2n) = 2n.
As a composition of related permutations:
a(n) = A255551(A255128(n)).
a(n) = A255553(A255408(n)).

A269355 Permutation of natural numbers: a(n) = A269380(A250469(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 23, 10, 11, 12, 13, 14, 15, 16, 9, 18, 17, 20, 31, 22, 25, 24, 21, 26, 27, 28, 19, 30, 29, 32, 49, 34, 71, 36, 37, 38, 39, 40, 41, 42, 43, 44, 107, 46, 47, 48, 119, 50, 51, 52, 35, 54, 89, 56, 101, 58, 53, 60, 61, 62, 63, 64, 115, 66, 67, 68, 173, 70, 55, 72, 33, 74, 75, 76, 131, 78, 77, 80, 167

Offset: 1

Antti Karttunen, Mar 03 2016

nonn

			For n=9 we first find what number is below 9 in square array A083221, which is 25, then we find what number is above 25 in square array A255127, which is 23, thus a(9) = 23.

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

Inverse: A269356.
Cf. A250469, A269380.
Cf. also arrays A083221 & A255127.
More recursed variant: A269357. Cf. also permutations A266645, A255407, A269171.

(define (A269355 n) (A269380 (A250469 n)))

a(n) = A269380(A250469(n)).
Other identities. For all n >= 1:
a(2n) = 2n. [Fixes the even numbers.]

A269356 Permutation of natural numbers: a(n) = A268674(A269379(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 17, 10, 11, 12, 13, 14, 15, 16, 19, 18, 29, 20, 25, 22, 9, 24, 23, 26, 27, 28, 31, 30, 21, 32, 73, 34, 53, 36, 37, 38, 39, 40, 41, 42, 43, 44, 107, 46, 47, 48, 33, 50, 51, 52, 59, 54, 71, 56, 137, 58, 101, 60, 61, 62, 63, 64, 109, 66, 67, 68, 121, 70, 35, 72, 97, 74, 75, 76, 79, 78, 131, 80, 197

Offset: 1

Antti Karttunen, Mar 03 2016

nonn

			For n=9 we first find what number is below 9 in square array A255127, which is 19, then we find what number is above 19 in square array A083221, which is 17, thus a(9) = 17.

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

Inverse: A269355.
Cf. A268674, A269379.
Cf. also arrays A083221 & A255127.
More recursed variant: A269358. Cf. also permutations A266646, A255408, A269172.

(define (A269356 n) (A268674 (A269379 n)))

a(n) = A268674(A269379(n)).
Other identities. For all n >= 1:
a(2n) = 2n. [Fixes the even numbers.]

A276620 Square array A(row,col) = A276610(row,col+1) - A276610(row,col): the first differences of each row of array A276610, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

4, 4, 8, 4, 10, 10, 4, 8, 12, 20, 4, 10, 12, 20, 16, 4, 8, 12, 22, 12, 24, 4, 10, 10, 24, 16, 26, 38, 4, 8, 10, 24, 14, 30, 40, 20, 4, 10, 14, 22, 14, 20, 38, 18, 34, 4, 8, 10, 16, 12, 32, 34, 14, 22, 40, 4, 10, 10, 22, 20, 26, 38, 22, 34, 68, 36, 4, 8, 12, 28, 14, 18, 40, 14, 26, 46, 36, 22, 4, 10, 12, 20, 8, 30, 48, 12, 30, 74, 28, 14, 34

Offset: 1

Antti Karttunen, Sep 13 2016

The first negative term occurs as a(490) = A(25,7) = A276610(25,8) - A276610(25,7) = 248 - 258 = -10.

			The top left 18 x 16 corner of the array:
   4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4
   8, 10,  8, 10,  8, 10,  8, 10,  8, 10,  8, 10,  8, 10,  8, 10,  8, 10
  10, 12, 12, 12, 10, 10, 14, 10, 10, 12, 12, 12, 10, 10, 14, 10, 10, 12
  20, 20, 22, 24, 24, 22, 16, 22, 28, 20, 22, 20, 20, 26, 20, 22, 24, 22
  16, 12, 16, 14, 14, 12, 20, 14,  8, 22, 12, 12, 18, 12, 12, 18, 14, 14
  24, 26, 30, 20, 32, 26, 18, 30, 28, 22, 28, 30, 28, 20, 30, 24, 28, 26
  38, 40, 38, 34, 38, 40, 48, 34, 32, 40, 40, 36, 40, 44, 36, 40, 36, 32
  20, 18, 14, 22, 14, 12, 20, 22, 20, 16, 18, 12, 20, 10, 24, 18, 18, 18
  34, 22, 34, 26, 30, 40, 20, 30, 30, 32, 28, 34, 26, 40, 26, 26, 30, 32
  40, 68, 46, 74, 52, 48, 70, 40, 64, 66, 46, 58, 58, 48, 58, 62, 58, 52
  36, 36, 28, 32, 30, 28, 18, 58, 28, 20, 38, 26, 30, 50, 30, 28, 22, 44
  22, 14, 14, 10, 46, 16, 24,  2, 32, 16, 24, 20, 18, 12, 28, 12, 30, 20
  34, 32, 46, 32, 16, 46, 30, 30, 30, 54, 34, 26, 34, 26, 30, 38, 40, 18
  52, 44, 40, 52, 40, 56, 58, 60, 28, 24, 64, 52, 54, 66, 30, 54, 30, 66
  52, 74, 70, 52, 66, 68, 56, 52, 80, 64, 42, 90, 54, 42, 78, 58, 72, 68
  42, 46, 58, 64, 38, 38, 62, 52, 44, 62, 52, 28, 62, 52, 50, 50, 42, 52

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array

Transpose: A276619.
Column 1: A276607.
Cf. A276610, A255127.

(define (A276620 n) (A276620bi (A002260 n) (A004736 n)))
(define (A276620bi row col) (- (A276610bi row (+ 1 col)) (A276610bi row col))) ;; Code for A276610bi given in A276610.

A(row,col) = A276610(row,col+1) - A276610(row,col).

A302035 a(1) = 0, for n > 1, a(n) = A001511(A260739(n)); Number of instances of (the smallest) Ludic factor A272565(n) in n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 01 2018

nonn

An A067029 analog for "Ludic factorization": iterating the map n -> A302034(n) until 1 is reached, and taking the Ludic factor (A272565) of each term gives a sequence of distinct Ludic numbers (A003309) in ascending order, while applying this function (A302035) to those terms gives the corresponding "exponents" of those Ludic factors, that is, the count of consecutive occurrences of each when iterating the map n -> A302032(n), which gives the same factors with repetitions. Permutation pair A302025/A302026 maps between the Ludic factorization and the ordinary prime factorization of n. See also comments and examples in A302032.

Antti Karttunen, Table of n, a(n) for n = 1..10105
Index entries for sequences generated by sieves

Cf. A001511, A003309, A255127, A260739, A302025, A302026, A302032, A302034.

Cf. also A067029, A302045.

a(1) = 0; for n > 1, a(n) = A001511(A260739(n)).

For n > 1, a(n) = A302025(A067029(A302026(n))).

cp's OEIS Frontend

A255408 Permutation of natural numbers: a(n) = A083221(A255128(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

Formula

A269171 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A269379(a(A268674(2n+1))).

Views

Author

Keywords

Links

Crossrefs

Formula

A260717 Square array: row n gives the numbers remaining before the stage n of Ludic sieve.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

A302036 Ludic powers: numbers k such that A302031(k) < 2; numbers k such that A260739(k) is a power of 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A255129 Transposed Ludic array.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

A260436 Permutation mapping from Ludic sieve to Lucky sieve: a(1) = 1, for n > 1: a(n) = A255551(A260738(n), A260739(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A269355 Permutation of natural numbers: a(n) = A269380(A250469(n)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Scheme

Formula

A269356 Permutation of natural numbers: a(n) = A268674(A269379(n)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Scheme

Formula

A276620 Square array A(row,col) = A276610(row,col+1) - A276610(row,col): the first differences of each row of array A276610, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Views

Author