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Search: a269171 -id:a269171
Displaying 1-9 of 9 results found. page 1
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A269393 Permutation of natural numbers: a(n) = A269171(3*n) / 3. +20
4
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 45, 38, 39, 40, 41, 42, 43, 44, 49, 46, 47, 48, 37, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 73, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 81, 90, 91, 76, 77, 78, 79, 80, 85 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The first non-fixed term is a(37)=45.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers
FORMULA
a(n) = A269171(3*n) / 3.
PROG
(Scheme) (define (A269393 n) (/ (A269171 (* 3 n)) 3))
CROSSREFS
Inverse: A269394.
Cf. A269171, A269395.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Mar 03 2016
STATUS
approved
A269395 Permutation of natural numbers: a(n) = A269171(A269393(n)) = A269171(A269171(3*n)/3). +20
4
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, 45, 46, 39, 40, 43, 42, 47, 44, 31, 50, 53, 48, 41, 38, 51, 52, 61, 54, 49, 56, 57, 58, 67, 60, 89, 74, 63, 64, 65, 66, 77, 68, 69, 70, 83, 72, 81, 90, 85, 92, 59, 78, 91, 80, 79 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Composition of A269171 with a permutation of natural numbers obtained from its trisection.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers
FORMULA
a(n) = A269171(A269393(n)) = A269171(A269171(3*n)/3).
PROG
(Scheme) (define (A269395 n) (A269171 (A269393 n)))
CROSSREFS
Inverse: A269396.
Cf. A269393.
Differs from A255407 and A269171 for the first time at n=37, which here a(37)=45, instead of 41.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Mar 03 2016
STATUS
approved
A269379 a(1) = 1; for n > 1, a(n) = A255127(A260738(n)+1, A260739(n)). +10
18
1, 3, 5, 9, 7, 15, 11, 21, 19, 27, 13, 33, 17, 39, 35, 45, 23, 51, 31, 57, 49, 63, 25, 69, 29, 75, 65, 81, 37, 87, 55, 93, 79, 99, 59, 105, 41, 111, 95, 117, 43, 123, 47, 129, 109, 135, 53, 141, 85, 147, 125, 153, 61, 159, 73, 165, 139, 171, 103, 177, 67, 183, 155, 189, 113, 195, 71, 201, 169, 207, 77, 213, 101, 219, 185, 225, 83 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(n) = the number located immediately below n in A255127 (square array generated by Ludic sieve) in the same column where n itself is, or in other words, the number removed in the next filtering stage at the same step as when n was removed in the A260738(n)-th stage.
Permutation of odd numbers.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..32999
FORMULA
a(1) = 1; for n > 1, a(n) = A255127(A260738(n)+1, A260739(n)).
Other identities. For all n >= 1:
A269380(a(n)) = n.
PROG
(Scheme) (define (A269379 n) (if (= 1 n) n (A255127bi (+ (A260738 n) 1) (A260739 n)))) ;; Code for A255127bi given in A255127.
CROSSREFS
Cf. A255127, A260738, A260739.
Cf. A269171, A269356, A269358, A269382, A269385, A269387 (sequences that use this function).
Cf. A269380 (left inverse).
Cf. also A250469, A269369.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Mar 01 2016
STATUS
approved
A269172 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A250469(a(A269380(2n+1))). +10
10
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, 50, 39, 40, 37, 42, 41, 44, 45, 38, 43, 48, 55, 46, 51, 52, 47, 54, 121, 56, 57, 58, 77, 60, 53, 98, 63, 64, 65, 66, 59, 68, 69, 70, 61, 72, 169, 62, 75, 100, 67, 78, 85, 80, 81 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..2244
Index entries for sequences that are permutations of the natural numbers
FORMULA
a(1) = 1, then after for even n, a(n) = 2*a(n/2), and for odd n, A250469(a(A269380(n))).
a(1) = 1, for n > 1, a(n) = A083221(A260738(n), a(A260739(n))).
As a composition of other permutations:
a(n) = A252755(A269386(n)).
a(n) = A252753(A269388(n)).
Other identities. For all n >= 1:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
a(A003309(n)) = A008578(n). [Maps Ludic numbers to noncomposites.]
PROG
(Scheme, two versions, both using memoization-macro definec)
(definec (A269172 n) (cond ((<= n 1) n) ((even? n) (* 2 (A269172 (/ n 2)))) (else (A250469 (A269172 (A269380 n))))))
(definec (A269172 n) (if (<= n 1) n (A083221bi (A260738 n) (A269172 (A260739 n))))) ;; Code for A083221bi given in A083221.
CROSSREFS
Inverse: A269171.
Cf. A000035, A003309, A008578, A083221, A250469, A260738, A260739, A269380.
Related or similar permutations: A260741, A260742, A269356, A269358, A255422.
Cf. also A269394 (a(3n)/3) and A269396.
Differs from A255408 for the first time at n=38, where a(38) = 50, while A255408(38) = 38.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Mar 03 2016
STATUS
approved
A302025 Permutation of natural numbers mapping ordinary factorization to "Ludic factorization": a(1) = 1, a(2n) = 2*a(n), a(A003961(n)) = A269379(a(n)). +10
9
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 23, 20, 27, 22, 25, 24, 19, 26, 21, 28, 29, 30, 37, 32, 39, 34, 35, 36, 41, 46, 63, 40, 43, 54, 47, 44, 33, 50, 53, 48, 31, 38, 75, 52, 61, 42, 65, 56, 99, 58, 67, 60, 71, 74, 57, 64, 95, 78, 77, 68, 135, 70, 83, 72, 89, 82, 51, 92, 59, 126, 91, 80, 45, 86, 97, 108, 155, 94, 147, 88 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
See comments and examples in A302032 to see how Ludic factorization proceeds.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences computed from indices in prime factorization
Index entries for sequences generated by sieves
FORMULA
a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A269379(a(A064989(2n+1))).
a(n) = A269171(A250245(n)).
a(n) = A269387(A156552(n)).
PROG
(PARI)
\\ With A269379 precomputed:
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)};
A302025(n) = if(1==n, n, if(!(n%2), 2*A302025(n/2), A269379(A302025(A064989(n)))));
(Scheme, with memoization-macro definec)
(definec (A302025 n) (cond ((= 1 n) n) ((even? n) (* 2 (A302025 (/ n 2)))) (else (A269379 (A302025 (A064989 n))))))
CROSSREFS
Cf. A302026 (inverse permutation).
Cf. A156552, A250245, A269171, A269387 (similar or related permutations).
Cf. A003961, A064989, A269379.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Apr 03 2018
STATUS
approved
A269387 Tree of Ludic sieve: a(0) = 1, a(1) = 2; after which, a(2n) = A269379(a(n)), a(2n+1) = 2*a(n). +10
8
1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 19, 18, 21, 16, 11, 14, 27, 20, 35, 30, 33, 24, 31, 38, 51, 36, 49, 42, 45, 32, 13, 22, 39, 28, 65, 54, 57, 40, 59, 70, 87, 60, 79, 66, 69, 48, 55, 62, 111, 76, 125, 102, 105, 72, 85, 98, 123, 84, 109, 90, 93, 64, 17, 26, 63, 44, 95, 78, 81, 56, 113, 130, 159, 108, 139, 114, 117, 80 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Permutation of natural numbers obtained from the Ludic sieve. Note the indexing: Domain starts from 0, range from 1.
This sequence can be represented as a binary tree. Each left hand child is obtained by applying A269379 to the parent's contents, and each right hand child is obtained by doubling the parent's contents:
1
|
...................2...................
3 4
5......../ \........6 9......../ \........8
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
7 10 15 12 19 18 21 16
11 14 27 20 35 30 33 24 31 38 51 36 49 42 45 32
etc.
Sequence A269385 is obtained from the mirror image of the same tree.
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers
FORMULA
a(0) = 1, a(1) = 2; after which, a(2n) = A269379(a(n)), a(2n+1) = 2*a(n).
As a composition of other permutations:
a(n) = A269171(A252753(n)).
a(n) = A260741(A269377(n)).
PROG
(Scheme, with memoization-macro definec)
(definec (A269387 n) (cond ((<= n 2) (+ 1 n)) ((even? n) (A269379 (A269387 (/ n 2)))) (else (* 2 (A269387 (/ (- n 1) 2))))))
CROSSREFS
Inverse: A269388.
Cf. A003309 (left edge of the tree).
Cf. A269379.
Related permutations: A260741, A269171, A269385.
Cf. also A252753, A269377.
KEYWORD
nonn,tabf
AUTHOR
Antti Karttunen, Mar 01 2016
STATUS
approved
A269385 Tree of Ludic sieve, mirrored: a(0) = 1, a(1) = 2; after which, a(2n) = 2*a(n), a(2n+1) = A269379(a(n)). +10
7
1, 2, 4, 3, 8, 9, 6, 5, 16, 21, 18, 19, 12, 15, 10, 7, 32, 45, 42, 49, 36, 51, 38, 31, 24, 33, 30, 35, 20, 27, 14, 11, 64, 93, 90, 109, 84, 123, 98, 85, 72, 105, 102, 125, 76, 111, 62, 55, 48, 69, 66, 79, 60, 87, 70, 59, 40, 57, 54, 65, 28, 39, 22, 13, 128, 189, 186, 229, 180, 267, 218, 191, 168, 249, 246, 305, 196, 291, 170, 151, 144 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Permutation of natural numbers obtained from the Ludic sieve. Note the indexing: Domain starts from 0, range from 1.
This sequence can be represented as a binary tree. Each left hand child is obtained by doubling the parent's contents, and each right hand child is obtained by applying A269379 to the parent's contents:
1
|
...................2...................
4 3
8......../ \........9 6......../ \........5
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
16 21 18 19 12 15 10 7
32 45 42 49 36 51 38 31 24 33 30 35 20 27 14 11
etc.
Sequence A269387 is obtained from the mirror image of the same tree.
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers
FORMULA
a(0) = 1, a(1) = 2; after which, a(2n) = 2*a(n), a(2n+1) = A269379(a(n)).
As a composition of related permutations:
a(n) = A269171(A252755(n)).
a(n) = A260741(A269375(n)).
Other identities. For all n >= 2:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n from a(2)=4 onward.]
PROG
(Scheme, with memoization-macro definec)
(definec (A269385 n) (cond ((<= n 1) (+ n 1)) ((even? n) (* 2 (A269385 (/ n 2)))) (else (A269379 (A269385 (/ (- n 1) 2))))))
CROSSREFS
Inverse: A269386.
Cf. A000035, A269379.
Cf. A003309 (right edge of the tree).
Related or similar permutations: A163511, A260741, A269387, A269171.
Cf. also A252755, A269375.
KEYWORD
nonn,tabf
AUTHOR
Antti Karttunen, Mar 01 2016
STATUS
approved
A269355 Permutation of natural numbers: a(n) = A269380(A250469(n)). +10
5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers
FORMULA
a(n) = A269380(A250469(n)).
Other identities. For all n >= 1:
a(2n) = 2n. [Fixes the even numbers.]
EXAMPLE
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.
PROG
(Scheme) (define (A269355 n) (A269380 (A250469 n)))
CROSSREFS
Inverse: A269356.
Cf. A250469, A269380.
Cf. also arrays A083221 & A255127.
More recursed variant: A269357. Cf. also permutations A266645, A255407, A269171.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Mar 03 2016
STATUS
approved
A269358 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A268674(A269379(2n+1)). +10
5
1, 2, 3, 4, 5, 6, 7, 8, 17, 10, 11, 12, 13, 14, 15, 16, 19, 34, 29, 20, 25, 22, 9, 24, 23, 26, 27, 28, 31, 30, 21, 32, 73, 38, 53, 68, 37, 58, 39, 40, 41, 50, 43, 44, 107, 18, 47, 48, 33, 46, 51, 52, 59, 54, 71, 56, 137, 62, 101, 60, 61, 42, 63, 64, 109, 146, 67, 76, 121, 106, 35, 136, 97, 74, 75, 116, 79, 78, 131, 80, 197 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
This is a variant of A269356, from which it differs for the first time at n=18.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers
FORMULA
a(1) = 1, after which, for even n, a(n) = 2*a(n/2) and for odd n, a(n) = A269356(n) = A268674(A269379(n)).
Other identities. For all n >= 1:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
PROG
(Scheme, with memoization-macro definec)
(definec (A269358 n) (cond ((<= n 1) n) ((even? n) (* 2 (A269358 (/ n 2)))) (else (A268674 (A269379 n)))))
CROSSREFS
Inverse: A269357.
Cf. A000035, A268674, A269379, A269356, A269171.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Mar 03 2016
STATUS
approved
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Last modified August 29 12:14 EDT 2024. Contains 375516 sequences. (Running on oeis4.)