A245705 -id:A245705 - cp's OEIS frontend

A245709 Fixed points of A245705 and A245706.

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 43, 48, 53, 64, 80, 86, 96, 106, 128, 160, 172, 192, 212, 249, 256, 320, 344, 384, 417, 424, 498, 512, 640, 688, 768, 834, 848, 996, 1024, 1280, 1321, 1376, 1536, 1668, 1696, 1992, 2048, 2560, 2642, 2752, 3072, 3336, 3392, 3984, 4096, 5120, 5284, 5504, 5545, 6144, 6672, 6784, 6827, 7081, 7968, 8192

Offset: 1

Antti Karttunen, Jul 30 2014

nonn

The odd terms less than 2^25 are: 1, 3, 5, 43, 53, 249, 417, 1321, 5545, 6827, 7081, 8535, 1485465, 1876261, 3298409, 13937375.

Contains also all such numbers k that A245608(k) = A245708(k), because that condition implies that A245607(A245708(k)) = k = A245707(A245608(k)). Conjecture: contains no numbers outside of that set, that is, for all n, A245608(a(n)) = A245708(a(n)).

Antti Karttunen, Table of n, a(n) for n = 1..230

A000079 is a subsequence.

Cf. A245705, A245706, A245608, A245708, A029747, A244154.

PARI
```
isA245709(n) = (A245705(n) == n);
```

\\ Alternative implementation, based on given conjecture:
isA245709v2(n) = (A245608(n) == A245708(n));

;; With Antti Karttunen's IntSeq-library.
(define A245709 (FIXED-POINTS 1 1 A245705))

;; With Antti Karttunen's IntSeq-library.
(define A245709 (MATCHING-POS 1 1 (lambda (k) (= (A245608 k) (A245708 k)))))

A245711 Permutation of natural numbers, odd bisection of A245705 incremented by one and halved: a(n) = (1+A245705((2*n)-1)) / 2.

Original entry on oeis.org

1, 2, 3, 5, 7, 4, 19, 10, 35, 9, 17, 139, 107, 6, 37, 36, 23, 13, 555, 229, 87, 22, 275, 1676, 39, 11, 27, 68, 63, 326, 137, 28, 4435, 18, 53, 147, 25, 8, 6819, 9388, 995, 16, 155, 73, 3659, 14, 533, 838, 145, 20, 49, 851, 29, 587, 535, 403, 4179, 26, 501, 1059, 1611, 1163, 69, 152, 663, 51, 283, 222, 181, 21

Offset: 1

Antti Karttunen, Jul 30 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: A245712.

Cf. A245705, A245605.

A245711(n) = (1+A245705((2*n)-1))/2;
for(n=1, 10001, write("b245711.txt", n, " ", A245711(n)));
\\ For the rest of code, see under A245705.

(define (A245711 n) (/ (+ 1 (A245705 (+ n n -1))) 2))

a(n) = (1+A245705((2*n)-1)) / 2.

A245605 Permutation of natural numbers: a(1) = 1, a(2n) = 2 * a(A064989(2n-1)), a(2n-1) = 1 + (2 * a(A064989(2n-1)-1)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 10, 7, 8, 13, 18, 17, 26, 11, 12, 37, 34, 25, 74, 15, 16, 69, 50, 21, 14, 19, 20, 33, 138, 41, 66, 35, 52, 53, 22, 277, 82, 31, 32, 45, 554, 65, 90, 27, 36, 1109, 130, 101, 42, 43, 28, 73, 2218, 149, 30, 71, 104, 57, 146, 209, 114, 51, 148, 133, 70, 293, 418, 555, 164, 141, 586, 329, 282, 75, 68, 105, 106, 1173, 658, 23, 24

Offset: 1

Antti Karttunen, Jul 29 2014

nonn

The even bisection halved gives A245607. The odd bisection incremented by one and halved gives A245707.

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

Inverse: A245606.
Cf. A048673, A064216, A064989, A245607, A245705, A245707.
Similar entanglement permutations: A244319, A005940, A163511, A243287, A243343, A243345, A244321, A245603, A245613, A135141, A237427.

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))));
for(n=1, 10001, write("b245605.txt", n, " ", A245605(n)));

;; With memoization-macro definec.
(definec (A245605 n) (cond ((= 1 n) 1) ((even? n) (* 2 (A245605 (A064989 (- n 1))))) (else (+ 1 (* 2 (A245605 (-1+ (A064989 n))))))))

a(1) = 1, a(2n) = 2 * a(A064989(2n-1)), a(2n-1) = 1 + (2 * a(A064989(2n-1)-1)).
a(1) = 1, a(2n) = 2 * a(A064216(n)), a(2n-1) = 1 + (2 * a(A064216(n)-1)).
As a composition of related permutations:
a(n) = A245607(A048673(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)).

A245706 Permutation of natural numbers: a(n) = A245707(A245608(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 11, 8, 7, 10, 27, 12, 9, 22, 75, 16, 19, 14, 15, 20, 51, 54, 659, 24, 35, 18, 91, 44, 565, 150, 83, 32, 21, 38, 67, 28, 13, 30, 99, 40, 139, 102, 43, 108, 33, 1318, 651, 48, 73, 70, 115, 36, 53, 182, 63, 88, 105, 1130, 19179, 300, 1107, 166, 8875, 64, 555, 42

Offset: 1

Antti Karttunen, Jul 30 2014

nonn

The even bisection halved gives the sequence back. The odd bisection incremented by one and halved gives A245712.

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

Inverse: A245705.
Fixed points: A245709.
Cf. A245608, A245707, A245606, A245712.

A245706(n) = A245707(A245608(n));
for(n=1, 10001, write("b245706.txt", n, " ", A245706(n)));
\\ Other code as in A245608 and A245707.

(define (A245706 n) (A245707 (A245608 n)))

a(n) = A245707(A245608(n)).

A245708 Permutation of natural numbers, the odd bisection of A245606 incremented by one and halved: a(n) = (1+A245606((2*n)-1))/2.

Original entry on oeis.org

1, 2, 3, 5, 4, 8, 6, 11, 7, 14, 13, 41, 10, 23, 63, 20, 15, 17, 9, 50, 16, 26, 21, 65, 45, 32, 18, 44, 30, 413, 58, 95, 22, 53, 12, 29, 27, 38, 66, 221, 52, 122, 48, 77, 115, 83, 748, 179, 69, 263, 25, 365, 39, 113, 153, 176, 130, 158, 508, 1007, 247, 140, 78, 242, 97, 59, 33, 89, 72, 68, 36, 47, 49, 188, 28

Offset: 1

Antti Karttunen, Jul 30 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: A245707.

Cf. A003961, A048673, A245606, A245608, A245705, A245709.

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))));
A245708(n) = (1+A245606((2*n)-1))/2;
for(n=1, 10001, write("b245708.txt", n, " ", A245708(n)))

(define (A245708 n) (* (/ 1 2) (+ 1 (A245606 (-1+ (* 2 n))))))

(define (A245708 n) (if (= 1 n) n (A048673 (+ 1 (A245606 (- n 1))))))

a(n) = (1+A245606((2*n)-1))/2.
As a composition of related permutations:
a(1) = 1, and for n > 1, a(n) = A048673(1+A245606(n-1)).
a(n) = A245608(A245705(n)).
Other identities:
For all n >= 0, a(2^n) = A245608(2^n). Moreover, A245709 gives all such k that a(k) = A245608(k).

cp's OEIS Frontend

A245709 Fixed points of A245705 and A245706.

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A245711 Permutation of natural numbers, odd bisection of A245705 incremented by one and halved: a(n) = (1+A245705((2*n)-1)) / 2.

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A245605 Permutation of natural numbers: a(1) = 1, a(2n) = 2 * a(A064989(2n-1)), a(2n-1) = 1 + (2 * a(A064989(2n-1)-1)).

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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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A245706 Permutation of natural numbers: a(n) = A245707(A245608(n)).

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A245708 Permutation of natural numbers, the odd bisection of A245606 incremented by one and halved: a(n) = (1+A245606((2*n)-1))/2.

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