A227413 -id:A227413 - cp's OEIS frontend

A243288 Permutation of natural numbers: a(1)=1, a(2n) = A102750(a(n)), a(2n+1) = A070003(a(n)).

1, 2, 4, 3, 8, 6, 16, 5, 9, 12, 32, 10, 25, 22, 81, 7, 18, 13, 36, 17, 54, 42, 242, 14, 49, 34, 150, 30, 128, 99, 882, 11, 27, 24, 100, 19, 64, 46, 256, 23, 98, 68, 490, 55, 338, 279, 4624, 20, 72, 62, 432, 44, 245, 178, 2209, 40, 216, 154, 1800, 119, 1200, 966

Offset: 1

Antti Karttunen, Jun 02 2014

nonn

This is an instance of "entanglement permutation", where two pairs of complementary subsets of natural numbers are interwoven with each other. In this case complementary pair odd/even numbers (A005408/A005843) is entangled with complementary pair A070003/A102750 (numbers which are divisible/not divisible by the square of their largest prime factor).

Thus this shares with the permutation A122111 the property that each even number is mapped to a unique term of A102750 and each odd number (larger than 1) to a unique term of A070003.

Antti Karttunen, Table of n, a(n) for n = 1..94
Index entries for sequences that are permutations of the natural numbers

Inverse of A243287.
Cf. A005843, A005408, A070003, A102750, A243282, A243285, A241917, A122111.
Similarly constructed permutations: A243343-A243346, A135141-A227413, A237126-A237427, A193231.

a(1)=1, and for n > 1, if n=2k, a(n) = A102750(a(k)), otherwise, when n = 2k+1, a(n) = A070003(a(k)).

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)).

A243346 a(1) = 1, a(2n) = A005117(1+a(n)), a(2n+1) = A013929(a(n)), where A005117 are squarefree and A013929 are nonsquarefree numbers.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 12, 5, 9, 13, 24, 10, 18, 19, 32, 7, 16, 14, 25, 21, 36, 38, 63, 15, 27, 30, 49, 31, 50, 53, 84, 11, 20, 26, 45, 22, 40, 39, 64, 34, 54, 59, 96, 62, 99, 103, 162, 23, 44, 42, 72, 47, 80, 79, 126, 51, 81, 82, 128, 86, 136, 138, 220, 17, 28, 33, 52, 41, 68, 73, 120

Offset: 1

Antti Karttunen, Jun 03 2014

nonn

This permutation entangles complementary pair A005843/A005408 (even/odd numbers) with complementary pair A005117/A013929 (numbers which are squarefree/are not squarefree).

Antti Karttunen, Table of n, a(n) for n = 1..8189
Index entries for sequences that are permutations of the natural numbers

Inverse: A243345.
Cf. A005843, A005408, A008966, A005117, A013929, A013928, A057627.
Similar permutations: A075378, A243343-A243344, A243347, A243287-A243288, A135141-A227413, A237126-A237427, A193231.

a(1) = 1, a(2n) = A005117(1+a(n)), a(2n+1) = A013929(a(n)).

For all n > 1, A008966(a(n)) = A000035(n+1), or equally, mu(a(n)) + 1 = n modulo 2, where mu is Moebius mu (A008683). [A property shared with a simpler variant A075378].

A232752 Permutation of natural numbers: a(0)=0, a(1)=1, a(2n)=A005228(1+(a(n))), a(2n+1)=A030124(a(n)).

Original entry on oeis.org

0, 1, 3, 2, 12, 5, 7, 4, 114, 16, 26, 8, 45, 10, 18, 6, 7562, 127, 191, 21, 462, 32, 56, 11, 1285, 53, 83, 14, 236, 23, 35, 9, 29172079, 7677, 9314, 141, 20528, 208, 312, 27, 115291, 489, 679, 39, 1943, 65, 98, 15, 865555, 1331, 1751, 62, 4111, 94, 150, 19, 30983, 255, 369, 29, 802, 42, 69, 13

Offset: 0

Antti Karttunen, Nov 30 2013

nonn

This is one example of the generic class of "entangling of two pairs of complementary sets" permutations of natural numbers. In this case, the Hofstadter's complementary pair A005228 & A030124 is entangled with complementary pair of A005843 & A005408, the even & odd numbers.
Note how, apart from 1, all the other terms of A005228 (1, 3, 7, 12, 18, 26, ...) occur in even positions, and all the terms of A030124 (2, 4, 5, 6, 8, 9, 10, 11, 13, 14, ...) occur in odd positions.
Moreover, at the positions given by two's powers, from 2^1 = 2 onwards, a(2^n) = 3, 12, 114, 7562, 29172079, ... the values are iterates of function b(n) = A005228(n+1) from b(1)=3 onward: b(1)=3, b(b(1))=12, b(b(b(1)))=114, b(b(b(b(1))))=7562, and so on.
In the same way, at the positions given by A000225, from 2^2 - 1 = 3 onwards, the iterates of A030124 appear, A030124(1), A030124(A030124(1)), A030124(A030124(A030124(1))), and so on, as: 2, 4, 6, 9, 13, 17, ... (= A232739).
The permutation A227413 is obtained in analogous way by entangling primes and composites with even and odd numbers.

Inverse permutation: A232751.
Cf. A227413, A005228, A030124, A005843 & A005408, A000079, A000225, A232739.
Cf. also the permutation pair A167151 & A225850.

a(0)=0, a(1)=1, and for even n > 1, a(n) = A005228(1+(a(n/2))), for odd n > 1, a(n) = A030124(a((n-1)/2)).

For all n >=1, a(A000225(n+1)) = A232739(n).

A243347 a(1)=1, and for n>1, if mu(n) = 0, a(n) = A005117(1+a(A057627(n))), otherwise, a(n) = A013929(a(A013928(n))).

Original entry on oeis.org

1, 4, 12, 2, 32, 8, 84, 6, 19, 24, 220, 3, 18, 50, 63, 53, 564, 13, 9, 138, 49, 128, 162, 10, 31, 136, 38, 365, 1448, 36, 25, 5, 351, 126, 332, 30, 414, 27, 81, 82, 348, 99, 931, 103, 86, 3699, 96, 929, 21, 14, 64, 223, 16, 79, 892, 210, 325, 847, 80, 265, 1056, 72, 15, 51, 208, 212, 884, 221, 256

Offset: 1

Antti Karttunen, Jun 03 2014

nonn

Self-inverse permutation of natural numbers.

Shares with A088609 the property that after 1, positions indexed by squarefree numbers larger than one, A005117(n+1): 2, 3, 5, 6, 7, 10, 11, 13, 14, ... contain only nonsquarefree numbers A013929: 4, 8, 9, 12, 16, 18, 20, 24, ..., and vice versa. However, instead of placing terms in those subsets in monotone order this sequence recursively permutes the order of both subsets with the emerging permutation itself, thus implementing a kind of "deep" variant of A088609. Alternatively, this can be viewed as yet another "entanglement permutation", where two pairs of complementary subsets of natural numbers are interwoven with each other. In this case complementary pair A005117/A013929 is entangled with complementary pair A013929/A005117.

Antti Karttunen, Table of n, a(n) for n = 1..520
Index entries for sequences that are permutations of the natural numbers

Cf. A008966, A005117, A013929, A013928, A057627, A088609, A243348.

Similar permutations: A236854, A235491, A243343-A243346, A243347, A243287-A243288, A135141-A227413, A237126-A237427, A193231.

a(1), and for n>1, if mu(n) = 0, a(n) = A005117(1+a(A057627(n))), otherwise, a(n) = A013929(a(A013928(n))). [Here mu is Moebius mu-function, A008683, which is zero only when n is a nonsquarefree number, one of the numbers in A013929.]

For all n > 1, A008966(a(n)) = 1 - A008966(n), or equally, mu(a(n)) + 1 = mu(n) modulo 2, where mu is Moebius mu (A008683). [Note: Permutation A088609 satisfies the same condition.]

A233280 Permutation of nonnegative integers: a(n) = A003188(A054429(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 11, 10, 14, 15, 13, 12, 16, 17, 19, 18, 22, 23, 21, 20, 28, 29, 31, 30, 26, 27, 25, 24, 32, 33, 35, 34, 38, 39, 37, 36, 44, 45, 47, 46, 42, 43, 41, 40, 56, 57, 59, 58, 62, 63, 61, 60, 52, 53, 55, 54, 50, 51, 49, 48, 64, 65, 67, 66

Offset: 0

Antti Karttunen, Dec 18 2013

nonn

This permutation transforms the enumeration system of positive irreducible fractions A071766/A229742 (HCS) into the enumeration system A007305/A047679 (Stern-Brocot), and the enumeration system A245325/A245326 into A162909/A162910 (Bird). - Yosu Yurramendi, Jun 09 2015

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A233279.
Cf. A000069, A001969, A003188, A054429, A063946, A154436.
Similarly constructed permutation pairs: A003188/A006068, A135141/A227413, A232751/A232752, A233275/A233276, A233277/A233278, A193231 (self-inverse).

from sympy import floor
def a003188(n): return n^(n>>1)
def a054429(n): return 1 if n==1 else 2*a054429(floor(n/2)) + 1 - n%2
def a(n): return 0 if n==0 else a003188(a054429(n)) # Indranil Ghosh, Jun 11 2017

maxrow <- 8 # by choice
a <- 1
for(m in 0:maxrow) for(k in 0:(2^m-1)){
a[2^(m+1)+    k] <- a[2^m+      k] + 2^m
a[2^(m+1)+2^m+k] <- a[2^(m+1)-1-k] + 2^(m+1)
}
a
# Yosu Yurramendi, Apr 05 2017

(define (A233280 n) (A003188 (A054429 n)))
;; Alternative version, based on entangling even & odd numbers with odious and evil numbers:
(definec (A233280 n) (cond ((< n 2) n) ((even? n) (A000069 (+ 1 (A233280 (/ n 2))))) (else (A001969 (+ 1 (A233280 (/ (- n 1) 2)))))))

a(n) = A003188(A054429(n)).
a(n) = A063946(A003188(n)).
a(n) = A054429(A154436(n)).
a(0)=0, a(1)=1, and otherwise, a(2n) = A000069(1+a(n)), a(2n+1) = A001969(1+a(n)). [A recurrence based on entangling even & odd numbers with odious and evil numbers]
a(n) = A258746(A180201(n)) = A180201(A117120(n)), n > 0. - Yosu Yurramendi, Apr 10 2017

A243344 a(1) = 1, a(2n) = A013929(a(n)), a(2n+1) = A005117(1+a(n)).

Original entry on oeis.org

1, 4, 2, 12, 6, 8, 3, 32, 19, 18, 10, 24, 13, 9, 5, 84, 53, 50, 31, 49, 30, 27, 15, 63, 38, 36, 21, 25, 14, 16, 7, 220, 138, 136, 86, 128, 82, 81, 51, 126, 79, 80, 47, 72, 42, 44, 23, 162, 103, 99, 62, 96, 59, 54, 34, 64, 39, 40, 22, 45, 26, 20, 11, 564, 365

Offset: 1

Antti Karttunen, Jun 03 2014

nonn

This permutation entangles complementary pair odd/even numbers (A005408/A005843) with complementary pair A005117/A013929 (numbers which are squarefree/not squarefree).

Antti Karttunen, Table of n, a(n) for n = 1..4095
Index entries for sequences that are permutations of the natural numbers

Inverse: A243343.
Cf. A005843, A005408, A008966, A005117, A013929, A013928, A057627.
Similar permutations: A088610 (simple variant), A243345-A243346, A243347, A243287-A243288, A135141-A227413, A237126-A237427, A193231.

a(1) = 1, a(2n) = A013929(a(n)), a(2n+1) = A005117(1+a(n)).

For all n, A008966(a(n)) = A000035(n), or equally, mu(a(n)) = n modulo 2, where mu is Moebius mu (A008683). [The same property holds for A088610.]

A243345 a(1)=1; thereafter, if n is k-th squarefree number [i.e., n = A005117(k)], a(n) = 2a(k-1); otherwise, when n is k-th nonsquarefree number [i.e., n = A013929(k)], a(n) = 2a(k)+1.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 16, 5, 9, 12, 32, 7, 10, 18, 24, 17, 64, 13, 14, 33, 20, 36, 48, 11, 19, 34, 25, 65, 128, 26, 28, 15, 66, 40, 72, 21, 96, 22, 38, 37, 68, 50, 130, 49, 35, 256, 52, 129, 27, 29, 56, 67, 30, 41, 132, 73, 80, 144, 42, 97, 192, 44, 23, 39, 76, 74, 136, 69, 100

Offset: 1

Antti Karttunen, Jun 03 2014

Any other fixed points than 1, 2, 6, 9, 135, 147, 914, ... ?

Any other points than 4, 21, 39, 839, 4893, 12884, ... where a(n) = n-1 ?

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A243346.
Cf. A005843, A005408, A008966, A005117, A013929, A013928, A057627.
Similar permutations: A243343-A243344, A243347, A243287-A243288, A135141-A227413, A237126-A237427, A193231.

a(1) = 1, and for n>1, if mu(n) = 0, a(n) = 1 + 2*a(A057627(n)), otherwise a(n) = 2*a(A013928(n)), where mu is Moebius mu function (A008683).

For all n > 1, A000035(a(n)+1) = A008966(n) = A008683(n)^2, or equally, a(n) = mu(n) + 1 modulo 2.

A244322 Permutation of natural numbers: a(1)=1, a(2n) = A244991(a(n)), a(2n+1) = A244990(1+a(n)).

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 7, 8, 9, 11, 13, 10, 12, 15, 14, 16, 18, 17, 19, 22, 24, 25, 27, 20, 21, 23, 26, 31, 29, 30, 28, 32, 35, 34, 37, 33, 36, 40, 38, 45, 43, 47, 49, 50, 52, 55, 54, 41, 39, 44, 42, 46, 48, 51, 53, 64, 61, 60, 57, 62, 58, 59, 56, 66, 63, 69, 71, 68, 70, 75

Offset: 1

Antti Karttunen, Jul 22 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: A244321.
Cf. A244990, A244991, A244992.
Similar entanglement permutations: A227413, A237126, A243288, A243344, A243346.

a(1)=1, a(2n) = A244991(a(n)), a(2n+1) = A244990(1+a(n)).

For all n >= 1, A244992(a(n)) = 1 - A000035(n).

A245702 Permutation of natural numbers: a(1) = 1, a(2n) = A014580(a(n)), a(2n+1) = A091242(a(n)), where A014580(n) = binary code for n-th irreducible polynomial over GF(2) and A091242(n) = binary code for n-th reducible polynomial over GF(2).

Original entry on oeis.org

1, 2, 4, 3, 5, 11, 8, 7, 6, 13, 9, 47, 17, 31, 14, 25, 12, 19, 10, 59, 20, 37, 15, 319, 62, 87, 24, 185, 42, 61, 21, 137, 34, 55, 18, 97, 27, 41, 16, 415, 76, 103, 28, 229, 49, 67, 22, 3053, 373, 433, 79, 647, 108, 131, 33, 1627, 222, 283, 54, 425, 78, 109, 29, 1123, 166, 203, 45, 379, 71, 91, 26, 731, 121, 145, 36, 253, 53, 73, 23

Offset: 1

Antti Karttunen, Aug 02 2014

nonn

Inverse: A245701.
Cf. A000035, A014580, A091242, A091225.
Similar entanglement permutations: A193231, A227413, A237126, A243288, A245703, A245704.

allocatemem(123456789);
a014580 = vector(2^18);
a091242 = vector(2^22);
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; j=0; n=2; while((n < 2^22), if(isA014580(n), i++; a014580[i] = n, j++; a091242[j] = n); n++)
A245702(n) = if(1==n, 1, if(0==(n%2), a014580[A245702(n/2)], a091242[A245702((n-1)/2)]));
for(n=1, 383, write("b245702.txt", n, " ", A245702(n)));

;; With memoizing definec-macro.
(definec (A245702 n) (cond ((< n 2) n) ((even? n) (A014580 (A245702 (/ n 2)))) (else (A091242 (A245702 (/ (- n 1) 2))))))

a(1) = 1, a(2n) = A014580(a(n)), a(2n+1) = A091242(a(n)).
As a composition of related permutations:
a(n) = A245703(A227413(n)).
Other identities:
For all n >= 1, 1 - A091225(a(n)) = A000035(n). [Maps even numbers to binary representations of irreducible GF(2) polynomials (= A014580) and odd numbers to the corresponding representations of reducible polynomials].

cp's OEIS Frontend

A243288 Permutation of natural numbers: a(1)=1, a(2n) = A102750(a(n)), a(2n+1) = A070003(a(n)).

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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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A243346 a(1) = 1, a(2n) = A005117(1+a(n)), a(2n+1) = A013929(a(n)), where A005117 are squarefree and A013929 are nonsquarefree numbers.

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A232752 Permutation of natural numbers: a(0)=0, a(1)=1, a(2n)=A005228(1+(a(n))), a(2n+1)=A030124(a(n)).

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A243347 a(1)=1, and for n>1, if mu(n) = 0, a(n) = A005117(1+a(A057627(n))), otherwise, a(n) = A013929(a(A013928(n))).

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A233280 Permutation of nonnegative integers: a(n) = A003188(A054429(n)).

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A243344 a(1) = 1, a(2n) = A013929(a(n)), a(2n+1) = A005117(1+a(n)).

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A243345 a(1)=1; thereafter, if n is k-th squarefree number [i.e., n = A005117(k)], a(n) = 2*a(k-1); otherwise, when n is k-th nonsquarefree number [i.e., n = A013929(k)], a(n) = 2*a(k)+1.

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A244322 Permutation of natural numbers: a(1)=1, a(2n) = A244991(a(n)), a(2n+1) = A244990(1+a(n)).

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A245702 Permutation of natural numbers: a(1) = 1, a(2n) = A014580(a(n)), a(2n+1) = A091242(a(n)), where A014580(n) = binary code for n-th irreducible polynomial over GF(2) and A091242(n) = binary code for n-th reducible polynomial over GF(2).

A243345 a(1)=1; thereafter, if n is k-th squarefree number [i.e., n = A005117(k)], a(n) = 2a(k-1); otherwise, when n is k-th nonsquarefree number [i.e., n = A013929(k)], a(n) = 2a(k)+1.