A269172 -id:A269172 - cp's OEIS frontend

A269394 Permutation of natural numbers: a(n) = A269172(3*n) / 3.

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, 49, 38, 39, 40, 41, 42, 43, 44, 37, 46, 47, 48, 45, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 97, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 61, 98, 99, 76, 77, 78, 79, 80, 73, 82, 83, 84, 81

Offset: 1

Antti Karttunen, Mar 03 2016

nonn

The first non-fixed term is a(37)=49.

Index entries for sequences that are permutations of the natural numbers

Inverse: A269393.

Cf. A269172, A269396.

(define (A269394 n) (/ (A269172 (* 3 n)) 3))

a(n) = A269172(3*n) / 3.

A269396 Permutation of natural numbers: a(n) = A269394(A269172(n)) = A269172(3*A269172(n)) / 3.

Original entry on oeis.org

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, 45, 32, 33, 34, 35, 36, 31, 50, 39, 40, 49, 42, 41, 44, 37, 38, 43, 48, 55, 46, 51, 52, 47, 54, 105, 56, 57, 58, 77, 60, 53, 90, 63, 64, 65, 66, 59, 68, 69, 70, 97, 72, 181, 62, 99, 100, 67, 78, 81, 80, 73

Offset: 1

Antti Karttunen, Mar 03 2016

nonn

Composition of A269172 with a permutation of natural numbers obtained from its trisection.

Index entries for sequences that are permutations of the natural numbers

Inverse: A269395.
Cf. A269394.
Differs from A255408 and A269172 for the first time at n=31, which here a(31)=45, instead of 49.

(define (A269396 n) (A269394 (A269172 n)))

a(n) = A269394(A269172(n)) = A269172(3*A269172(n)) / 3.

A269380 a(1) = 1, after which, for odd numbers: a(n) = A260739(n)-th number k for which A260738(k) = A260738(n)-1, and for even numbers: a(n) = a(n/2).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 11, 5, 6, 1, 13, 4, 9, 3, 8, 7, 17, 2, 23, 11, 10, 5, 25, 6, 19, 1, 12, 13, 15, 4, 29, 9, 14, 3, 37, 8, 41, 7, 16, 17, 43, 2, 21, 23, 18, 11, 47, 10, 31, 5, 20, 25, 35, 6, 53, 19, 22, 1, 27, 12, 61, 13, 24, 15, 67, 4, 55, 29, 26, 9, 71, 14, 33, 3, 28, 37, 77, 8, 49, 41, 30, 7, 83, 16, 89, 17, 32, 43, 39, 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 A255127 (square array generated by Ludic sieve), in the same column where n itself is.

Antti Karttunen, Table of n, a(n) for n = 1..33011

Cf. A255127, A260738, A260739.
Cf. A269172, A269355, A269357, A269382, A269386, A269388 (sequences that use this function).
Cf. also A268674, A269370.

a(1) = 1; after which, for even numbers a(n) = a(n/2), and for odd numbers a(n) = A255127(A260738(n)-1, A260739(n)).
Other identities. For all n >= 1:
a(A269379(n)) = n.

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.]

A269388 Permutation of natural numbers: a(1) = 0, after which, a(2n) = 1 + 2a(n), a(2n+1) = 2 a(A269380(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 11, 32, 17, 10, 15, 64, 13, 12, 19, 14, 33, 128, 23, 256, 65, 18, 35, 512, 21, 24, 31, 22, 129, 20, 27, 1024, 25, 34, 39, 2048, 29, 4096, 67, 30, 257, 8192, 47, 28, 513, 26, 131, 16384, 37, 48, 71, 38, 1025, 40, 43, 32768, 49, 66, 63, 36, 45, 65536, 259, 46, 41, 131072, 55, 96, 2049, 130, 51, 262144, 69, 44

Offset: 1

Antti Karttunen, Mar 01 2016

nonn

Note the indexing: Domain starts from 1, range from 0.

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

Inverse: A269387.
Cf. A269380.
Related permutations: A260742, A269386, A269172.
Cf. also A252754, A269378.
Differs from A156552, A252754 and A246677(n-1) for the first time at n=19, which here a(19)=12, instead of 128.

a(1) = 0, after which, a(2n) = 1 + 2*a(n), a(2n+1) = 2 * a(A269380(n)).
As a composition of other permutations:
a(n) = A252754(A269172(n)).
a(n) = A269378(A260742(n)).

A302026 Permutation of natural numbers mapping "Ludic factorization" to ordinary factorization: a(1) = 1, a(2n) = 2*a(n), a(A269379(n)) = A003961(a(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 25, 20, 27, 22, 19, 24, 23, 26, 21, 28, 29, 30, 49, 32, 45, 34, 35, 36, 31, 50, 33, 40, 37, 54, 41, 44, 81, 38, 43, 48, 125, 46, 75, 52, 47, 42, 121, 56, 63, 58, 77, 60, 53, 98, 39, 64, 55, 90, 59, 68, 135, 70, 61, 72, 169, 62, 51, 100, 67, 66, 175, 80, 99, 74, 71, 108, 343, 82, 105, 88

Offset: 1

Antti Karttunen, Apr 03 2018

nonn

See comments and examples in A302032 to see how Ludic factorization proceeds.

Cf. A302025 (inverse permutation).
Cf. A005940, A250246, A269172, A269388 (similar or related permutations).
Cf. A003961, A064989, A269379, A269380, A302031, A302032, A302034, A302037.

\\ With A269380 precomputed:
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A302026(n) = if(1==n,n,if(!(n%2),2*A302026(n/2),A003961(A302026(A269380(n)))));
(Scheme, with memoization-macro definec)
(definec (A302026 n) (cond ((= 1 n) n) ((even? n) (* 2 (A302026 (/ n 2)))) (else (A003961 (A302026 (A269380 n))))))

a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A003961(a(A269380(2n+1))).
a(n) = A250246(A269172(n)).
a(n) = A005940(1+A269388(n)).
Other identities. For all n >= 1:
A001221(a(n)) = A302031(n).
A001222(a(n)) = A302037(n).

A269386 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2a(n), a(2n+1) = 1 + 2a(A269380(2n+1)).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 15, 4, 5, 14, 31, 12, 63, 30, 13, 8, 127, 10, 11, 28, 9, 62, 255, 24, 511, 126, 29, 60, 1023, 26, 23, 16, 25, 254, 27, 20, 2047, 22, 61, 56, 4095, 18, 8191, 124, 17, 510, 16383, 48, 19, 1022, 21, 252, 32767, 58, 47, 120, 57, 2046, 55, 52, 65535, 46, 125, 32, 59, 50, 131071, 508, 49, 54, 262143, 40, 95, 4094, 253, 44

Offset: 1

Antti Karttunen, Mar 01 2016

nonn

Note the indexing: Domain starts from 1, range from 0.

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

Inverse: A269385.
Cf. A269380.
Related permutations: A260742, A269172, A269388.
Cf. also A252756, A269376.
Differs from A243071 and A252756 for the first time at n=19, which here a(19) = 11, instead of 255.

a(1) = 0, a(2) = 1, a(2n) = 2*a(n), a(2n+1) = 1 + 2*a(A269380(2n+1)).
As a composition of related permutations:
a(n) = A252756(A269172(n)).
a(n) = A269376(A260742(n)).

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.]

A269357 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A269380(A250469(2n+1)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 23, 10, 11, 12, 13, 14, 15, 16, 9, 46, 17, 20, 31, 22, 25, 24, 21, 26, 27, 28, 19, 30, 29, 32, 49, 18, 71, 92, 37, 34, 39, 40, 41, 62, 43, 44, 107, 50, 47, 48, 119, 42, 51, 52, 35, 54, 89, 56, 101, 38, 53, 60, 61, 58, 63, 64, 115, 98, 67, 36, 173, 142, 55, 184, 33, 74, 75, 68, 131, 78, 77, 80, 167

Offset: 1

Antti Karttunen, Mar 03 2016

nonn

This is a variant of A269355, from which it differs for the first time at n=18.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences that are permutations of the natural numbers

Inverse: A269358.

Cf. A000035, A250469, A269380, A269355, A269172.

a(1) = 1, after which, for even n, a(n) = 2*a(n/2) and for odd n, a(n) = A269355(n) = A269380(A250469(n)).
Other identities. For all n >= 1:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

cp's OEIS Frontend

A269394 Permutation of natural numbers: a(n) = A269172(3*n) / 3.

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A269396 Permutation of natural numbers: a(n) = A269394(A269172(n)) = A269172(3*A269172(n)) / 3.

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A269380 a(1) = 1, after which, for odd numbers: a(n) = A260739(n)-th number k for which A260738(k) = A260738(n)-1, and for even numbers: a(n) = a(n/2).

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

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A269388 Permutation of natural numbers: a(1) = 0, after which, a(2n) = 1 + 2*a(n), a(2n+1) = 2 * a(A269380(n)).

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A302026 Permutation of natural numbers mapping "Ludic factorization" to ordinary factorization: a(1) = 1, a(2n) = 2*a(n), a(A269379(n)) = A003961(a(n)).

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A269386 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2*a(n), a(2n+1) = 1 + 2*a(A269380(2n+1)).

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

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A269357 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A269380(A250469(2n+1)).

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A269388 Permutation of natural numbers: a(1) = 0, after which, a(2n) = 1 + 2a(n), a(2n+1) = 2 a(A269380(n)).

A269386 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2a(n), a(2n+1) = 1 + 2a(A269380(2n+1)).