A245706 -id:A245706 - 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)))))

A245712 Permutation of natural numbers, odd bisection of A245706 incremented by one and halved: a(n) = (1+A245706((2*n)-1)) / 2.

Original entry on oeis.org

1, 2, 3, 6, 4, 14, 5, 38, 10, 8, 26, 330, 18, 46, 283, 42, 11, 34, 7, 50, 70, 22, 17, 326, 37, 58, 27, 32, 53, 9590, 554, 4438, 278, 174, 9, 16, 15, 142, 25, 226, 146, 658, 105, 310, 1190, 806, 1770, 524, 51, 5258, 66, 5302, 35, 518, 17739, 1054, 19030, 542, 5550, 8054, 35498, 4434, 29, 810, 1106, 72, 75, 28, 63, 166, 83, 150, 44

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: A245711.

Cf. A245706, A245606.

A245712(n) =(1+A245706((2*n)-1))/2;
for(n=1, 10001, write("b245712.txt", n, " ", A245712(n)));
\\ For the rest of code, see under A245706.

(define (A245712 n) (/ (+ 1 (A245706 (+ n n -1))) 2))

a(n) = (1+A245706((2*n)-1)) / 2.

A245606 Permutation of natural numbers: a(1) = 1, a(2n) = 1 + A003961(a(n)), a(2n+1) = A003961(1+a(n)). [Where A003961(n) shifts the prime factorization of n one step left].

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 10, 7, 8, 15, 16, 11, 26, 21, 22, 13, 12, 27, 28, 25, 36, 81, 82, 19, 14, 45, 52, 125, 56, 39, 40, 29, 18, 33, 46, 17, 126, 99, 100, 31, 50, 51, 226, 41, 626, 129, 130, 89, 24, 63, 34, 35, 176, 87, 154, 59, 344, 825, 298, 115, 86, 189, 190, 43, 32, 105, 76, 23, 66, 57, 88, 53, 20

Offset: 1

Antti Karttunen, Jul 29 2014

nonn

The even bisection halved gives A245608. The odd bisection incremented by one and halved gives A245708.

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

Inverse: A245605.
Cf. A003961, A243501, A064216, A245608, A245706, A245708.
Similar entanglement permutations: A244319, A156552, A243071, A243288, A243344, A243346, A244322, A245604, A245614, A227413, A237126.

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))))

;; With memoization-macro definec.
(definec (A245606 n) (cond ((= 1 n) 1) ((even? n) (A243501 (A245606 (/ n 2)))) (else (A003961 (+ 1 (A245606 (/ (- n 1) 2)))))))

a(1) = 1, a(2n) = A243501(a(n)), a(2n+1) = A003961(1+a(n)).
As a composition of related permutations:
a(n) = A064216(A245608(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).

A245705 Permutation of natural numbers: a(n) = A245607(A245708(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 13, 10, 7, 12, 37, 18, 19, 16, 69, 26, 17, 20, 33, 14, 277, 24, 213, 74, 11, 36, 73, 38, 71, 32, 45, 138, 25, 52, 1109, 34, 457, 40, 173, 66, 43, 28, 549, 554, 3351, 48, 77, 426, 21, 148, 53, 22, 135, 72, 125, 146, 651, 76, 273, 142, 55, 64

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 A245711.

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

Inverse: A245706.
Fixed points: A245709.
Cf. A245607, A245708, A245605, A245711.

A245705(n) = A245607(A245708(n));
for(n=1, 10001, write("b245705.txt", n, " ", A245705(n)));
\\ Other code as in A245607 and A245708.

(define (A245705 n) (A245607 (A245708 n)))

a(n) = A245607(A245708(n)).

A245707 Permutation of natural numbers, the odd bisection of A245605 incremented by one and halved: a(n) = (1+A245605((2*n)-1))/2.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 9, 6, 19, 13, 8, 35, 11, 10, 17, 21, 18, 27, 139, 16, 23, 33, 14, 555, 51, 22, 37, 75, 36, 29, 105, 26, 67, 147, 278, 71, 165, 38, 53, 587, 12, 107, 83, 28, 25, 2219, 72, 43, 73, 20, 87, 41, 34, 291, 277, 210, 163, 31, 66, 149, 131, 330, 15, 229, 24, 39, 2347, 70, 49, 101

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: A245708.

Cf. A064216, A064989, A245605, A245607, A245706.

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))));
A245707(n) = (1+A245605((2*n)-1))/2;

(define (A245707 n) (* (/ 1 2) (+ 1 (A245605 (-1+ (* 2 n))))))

(define (A245707 n) (if (= 1 n) n (+ 1 (A245605 (-1+ (A064216 n))))))

a(n) = (1+A245605((2*n)-1))/2.
As a composition of related permutations:
a(1) = 1, and for n > 1, a(n) = 1+A245605(A064216(n)-1).
a(n) = A245706(A245607(n)).

cp's OEIS Frontend

A245709 Fixed points of A245705 and A245706.

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A245712 Permutation of natural numbers, odd bisection of A245706 incremented by one and halved: a(n) = (1+A245706((2*n)-1)) / 2.

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A245606 Permutation of natural numbers: a(1) = 1, a(2n) = 1 + A003961(a(n)), a(2n+1) = A003961(1+a(n)). [Where A003961(n) shifts the prime factorization of n one step left].

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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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A245705 Permutation of natural numbers: a(n) = A245607(A245708(n)).

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A245707 Permutation of natural numbers, the odd bisection of A245605 incremented by one and halved: a(n) = (1+A245605((2*n)-1))/2.

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