A245609 -id:A245609 - cp's OEIS frontend

A245611 Permutation of natural numbers: a(n) = A243071(A064216(n)).

0, 1, 3, 7, 2, 15, 31, 6, 63, 127, 14, 255, 5, 4, 511, 1023, 30, 13, 2047, 62, 4095, 8191, 12, 16383, 11, 126, 32767, 29, 254, 65535, 131071, 28, 61, 262143, 510, 524287, 1048575, 10, 27, 2097151, 8, 4194303, 125, 1022, 8388607, 59, 2046, 253, 16777215, 60, 33554431, 67108863, 26

Offset: 1

Antti Karttunen, Jul 28 2014

nonn

Note the indexing: the domain starts from 1, while the range includes also zero.

The odd bisection of A243071 decremented by one and halved. (For a(1) = 0, take ceiling of -1/2).

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

Inverse: A245612.

Cf. A054429, A064216, A243071, A243065-A243066, A243505-A243506, A245607, A245609, A244153, A244154.

(define (A245611 n) (A243071 (A064216 n))) ;; offset 1, a(1) = 0.

a(1) = 0, and for n > 1, a(n) = (1/2) * (A243071((2*n)-1) - 1).
As a composition of related permutations:
a(n) = A243071(A064216(n)).
a(n) = A054429(A244153(n)).

A244153 Permutation of natural numbers, the odd bisection of A156552 halved; equally, a composition of A064216 and A156552: a(n) = A156552(A064216(n)).

Original entry on oeis.org

0, 1, 2, 4, 3, 8, 16, 5, 32, 64, 9, 128, 6, 7, 256, 512, 17, 10, 1024, 33, 2048, 4096, 11, 8192, 12, 65, 16384, 18, 129, 32768, 65536, 19, 34, 131072, 257, 262144, 524288, 13, 20, 1048576, 15, 2097152, 66, 513, 4194304, 36, 1025, 130, 8388608, 35, 16777216, 33554432, 21, 67108864, 134217728, 2049, 268435456, 258, 67, 68, 24, 4097, 14

Offset: 1

Antti Karttunen, Jun 27 2014

nonn

Note the indexing: the domain starts from 1, while the range includes also zero.

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

Inverse: A244154.

Cf. A054429, A064216, A156552, A243065-A243066, A243505-A243506, A245607, A245609, A245611.

(define (A244153 n) (A156552 (A064216 n)))

a(n) = A156552(2n+1) / 2.
As a composition of related permutations:
a(n) = A156552(A064216(n)).
a(n) = A054429(A245611(n)).

A245607 Permutation of natural numbers, the even bisection of A245605 halved: a(n) = A245605(2*n)/2.

Original entry on oeis.org

1, 2, 3, 5, 4, 9, 13, 6, 17, 37, 8, 25, 7, 10, 69, 33, 26, 11, 41, 16, 277, 45, 18, 65, 21, 14, 1109, 15, 52, 73, 57, 74, 35, 209, 82, 293, 141, 34, 53, 329, 12, 1173, 31, 36, 213, 149, 104, 43, 49, 20, 145, 173, 138, 81, 581, 114, 553, 71, 90, 133, 101, 282, 19, 325, 24, 457, 165, 50, 77, 97, 62, 105, 555, 42

Offset: 1

Antti Karttunen, Jul 29 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245608.

Cf. A064216, A064989, A245705, A245707, A245605, A245609-A245610, A245611, A244153, A243065-A243066.

A064989(n) = my(f = factor(n)); for(i=1, #f~, if((2 == f[i,1]),f[i,1] = 1,f[i,1] = precprime(f[i,1]-1))); factorback(f);
A245605(n) = if(1==n, 1, if(0==(n%2), 2*A245605(A064989(n-1)), 1+(2*A245605(A064989(n)-1))));
A245607(n) = A245605(2*n)/2;
for(n=1, 10001, write("b245607.txt", n, " ", A245607(n)));

(define (A245607 n) (A245605 (A064216 n)))

a(n) = A245605(2*n)/2.
As a composition of related permutations:
a(n) = A245605(A064216(n)).
a(n) = A245705(A245707(n)).

A245608 Permutation of natural numbers, the even bisection of A245606 halved: a(n) = A245606(2*n)/2.

Original entry on oeis.org

1, 2, 3, 5, 4, 8, 13, 11, 6, 14, 18, 41, 7, 26, 28, 20, 9, 23, 63, 50, 25, 113, 313, 65, 12, 17, 88, 77, 172, 149, 43, 95, 16, 38, 33, 44, 10, 413, 163, 221, 19, 74, 48, 191, 22, 476, 118, 179, 49, 68, 138, 29, 39, 527, 78, 215, 31, 635, 1593, 227, 102, 71, 688, 242, 24, 122, 193, 104, 15, 98, 58, 176, 30, 32, 123

Offset: 1

Antti Karttunen, Jul 29 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245607.

Cf. A003961, A048673, A245606, A245609-A245610, A245706, A245708, A245709, A245612, A244154, A243065-A243066.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ This function from Michel Marcus
A245606(n) = if(1==n, 1, if(0==(n%2), 1+A003961(A245606(n/2)),A003961(1+A245606((n-1)/2))))
A245608(n) = A245606(2*n)/2;
for(n=1, 10001, write("b245608.txt", n, " ", A245608(n)))

(define (A245608 n) (A048673 (A245606 n)))

a(n) = A245606(2*n)/2.
As a composition of related permutations:
a(n) = A048673(A245606(n)).
a(n) = A245708(A245706(n)).
Other identities:
For all n >= 0, a(2^n) = A245708(2^n). Moreover, A245709 gives all such k that a(k) = A245708(k).

A244319 Self-inverse permutation of natural numbers: a(1) = 1, a(2n) = A003961(1+a(A064989(2n-1))), a(2n+1) = 1+A003961(a(A064989(2n+1)-1)).

Original entry on oeis.org

1, 3, 2, 9, 6, 5, 26, 11, 4, 21, 8, 125, 56, 25, 16, 15, 344, 115, 36, 1015, 10, 39, 204, 41, 14, 7, 52, 45, 86, 301, 176, 155, 298, 51, 50, 19, 518, 305, 22, 189, 24, 895, 1376, 49, 28, 825, 1268, 11875, 44, 35, 34, 27, 3186, 6625, 2388, 13, 454, 153, 126, 3191, 476, 131

Offset: 1

Antti Karttunen, Jul 18 2014; description corrected and PARI code added Jul 30 2014

nonn

After 1, maps each even number to a unique odd number and vice versa, i.e., for all n > 1, A000035(a(n)) XOR A000035(n) = 1, where XOR is given in A003987.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A003987, A003961, A064989, A243501.
Related permutations: A048673, A064216, A245609-A245610.
Similar entanglement permutations: A245605-A245606, A235491, A236854, A243347, A244152.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ This function from Michel Marcus
A064989(n) = my(f = factor(n)); for(i=1, #f~, if((2 == f[i,1]),f[i,1] = 1,f[i,1] = precprime(f[i,1]-1))); factorback(f);
A244319(n) = if(1==n, 1, if(0==(n%2), A003961(1+A244319(A064989(n-1))), 1+A003961(A244319(A064989(n)-1))));
for(n=1, 10001, write("b244319.txt", n, " ", A244319(n)))
(Scheme, with Antti Karttunen's IntSeq-library for memoizing definec-macro)
(definec (A244319 n) (cond ((= 1 n) 1) ((even? n) (A003961 (+ 1 (A244319 (A064989 (- n 1)))))) (else (A243501 (A244319 (-1+ (A064989 n)))))))

a(1) = 1, a(2n) = A003961(1+a(A064989(2n-1))), a(2n+1) = A243501(a(A064989(2n+1)-1)).
As a composition of related permutations:
a(n) = A245609(A048673(n)) = A064216(A245610(n)).

A245610 Permutation of natural numbers: a(n) = A048673(A244319(n)).

Original entry on oeis.org

1, 3, 2, 13, 8, 4, 26, 7, 5, 28, 14, 172, 149, 25, 41, 18, 635, 102, 113, 1194, 11, 43, 428, 22, 17, 6, 77, 88, 71, 259, 527, 130, 227, 48, 74, 12, 677, 235, 20, 688, 68, 634, 5711, 61, 50, 1593, 1490, 27612, 59, 39, 29, 63, 11438, 10119, 4748, 9, 344, 238, 413, 1602, 941, 69

Offset: 1

Antti Karttunen, Jul 29 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245609.

Cf. A003961, A048673, A244319, A245607-A245608.

A048673(n) = (1+A003961(n))/2;
A245610(n) = A048673(A244319(n)); \\ For the other needed functions, see under A244319.
for(n=1, 10001, write("b245610.txt", n, " ", A245610(n)));

(define (A245610 n) (A048673 (A244319 n)))

a(n) = A048673(A244319(n)).

cp's OEIS Frontend

A245611 Permutation of natural numbers: a(n) = A243071(A064216(n)).

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A244153 Permutation of natural numbers, the odd bisection of A156552 halved; equally, a composition of A064216 and A156552: a(n) = A156552(A064216(n)).

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A245607 Permutation of natural numbers, the even bisection of A245605 halved: a(n) = A245605(2*n)/2.

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A245608 Permutation of natural numbers, the even bisection of A245606 halved: a(n) = A245606(2*n)/2.

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A244319 Self-inverse permutation of natural numbers: a(1) = 1, a(2n) = A003961(1+a(A064989(2n-1))), a(2n+1) = 1+A003961(a(A064989(2n+1)-1)).

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A245610 Permutation of natural numbers: a(n) = A048673(A244319(n)).

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