A243287 -id:A243287 - cp's OEIS frontend

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.

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.

A245701 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = 2a(n), a(A091242(n)) = 2a(n)+1, where A014580(n) = binary code for n-th irreducible polynomial over GF(2), A091242(n) = binary code for n-th reducible polynomial over GF(2).

Original entry on oeis.org

1, 2, 4, 3, 5, 9, 8, 7, 11, 19, 6, 17, 10, 15, 23, 39, 13, 35, 18, 21, 31, 47, 79, 27, 16, 71, 37, 43, 63, 95, 14, 159, 55, 33, 143, 75, 22, 87, 127, 191, 38, 29, 319, 111, 67, 287, 12, 151, 45, 175, 255, 383, 77, 59, 34, 639, 223, 135, 20, 575, 30, 25, 303, 91, 351, 511, 46, 767, 155, 119, 69, 1279, 78, 447, 271, 41, 1151, 61, 51

Offset: 1

Antti Karttunen, Aug 02 2014

Antti Karttunen, Table of n, a(n) for n = 1..10001
Antti Karttunen, Entanglement Permutations, 2016-2017
Index entries for sequences related to polynomials in ring GF(2)[X]
Index entries for sequences that are permutations of the natural numbers

Inverse: A245702.
Cf. A000035, A014580, A091242, A091225, A091226, A091245.
Similar entanglement permutations: A135141, A193231, A237427, A243287, A245703, A245704.

allocatemem(123456789);
A091226 = vector(2^22);
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
n=2; while((n < 2^22), if(isA014580(n), A091226[n] = A091226[n-1]+1, A091226[n] = A091226[n-1]); n++)
A091245(n) = ((n-A091226[n])-1);
A245701(n) = if(1==n, 1, if(isA014580(n), 2*(A245701(A091226[n])), 1 + 2*(A245701(A091245(n)))));
for(n=1, 10001, write("b245701.txt", n, " ", A245701(n)));

;; With memoizing definec-macro.
(definec (A245701 n) (cond ((= 1 n) n) ((= 1 (A091225 n)) (* 2 (A245701 (A091226 n)))) (else (+ 1 (* 2 (A245701 (A091245 n)))))))

a(1) = 1, and for n > 1, if n is in A014580, a(n) = 2*a(A091226(n)), otherwise a(n) = 1 + 2*a(A091245(n)).
As a composition of related permutations:
a(n) = A135141(A245704(n)).
Other identities:
For all n >= 1, 1 - A000035(a(n)) = A091225(n). [Maps binary representations of irreducible GF(2) polynomials (= A014580) to even numbers and the corresponding representations of reducible polynomials to odd numbers].

A245613 Permutation of natural numbers: a(1) = 1; thereafter, if n is k-th number with an odd number of prime divisors (counted with multiplicity) [i.e., n = A026424(k)], a(n) = A244991(a(k)), otherwise, when n is k-th number > 1 with an even number of prime divisors [i.e., n = A028260(1+k)], a(n) = A244990(1+a(k)).

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 5, 16, 9, 7, 11, 10, 32, 18, 13, 12, 17, 15, 22, 20, 35, 19, 66, 14, 24, 21, 34, 25, 23, 33, 31, 45, 63, 37, 27, 26, 41, 36, 29, 43, 69, 40, 134, 30, 47, 39, 44, 68, 71, 50, 38, 46, 67, 131, 28, 49, 42, 70, 64, 52, 92, 48, 127, 65, 61, 75, 55, 51, 89, 83, 73, 60

Offset: 1

Antti Karttunen, Jul 27 2014

nonn

This shares with the permutation A122111 the property that each term of A028260 is mapped to a unique term of A244990 and each term of A026424 is mapped to a unique term of A244991.

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

Inverse: A245614.
Cf. A026424, A028260, A055037, A055038, A066829, A244990, A244991, A244992, A122111.
Similar entanglement permutations: A243287, A243343, A243345, A244321-A244322, A245603, A135141, A237427, A245605.

a(1) = 1, and for n > 1, if A066829(n) = 1, a(n) = A244991(a(A055038(n))), otherwise a(n) = A244990(1+a(A055037(n)-1)).
As a composition of related permutations:
a(n) = A244322(A245603(n)).
For all n >= 1, A066829(n) = A244992(a(n)).

A252757 Permutation of natural numbers: a(1)=1, and for n>1, if n is k-th number whose largest prime factor is less than the square of its smallest prime factor [i.e., n = A251726(k)], a(n) = 2a(k), otherwise, when n = A251727(k), a(n) = 1 + 2a(k).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 3, 512, 6, 1024, 5, 12, 2048, 10, 24, 4096, 9, 20, 17, 48, 8192, 18, 33, 40, 65, 34, 129, 96, 16384, 257, 513, 36, 66, 80, 7, 1025, 13, 130, 2049, 68, 11, 258, 25, 192, 32768, 514, 4097, 21, 49, 1026, 72, 132, 8193, 19, 41, 160, 35, 14, 97, 2050, 26, 260, 16385, 4098, 37, 67, 81, 136, 22

Offset: 1

Antti Karttunen, Jan 02 2015

nonn

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

Inverse: A252758.
Similar permutations: A243287, A135141, A237427.
Cf. A252372, A252373, A251726, A251727.

a(1)=1, and for n>1: if A252372(n) = 1 [i.e. the largest prime factor of n is less than the square of its smallest prime factor], a(n) = 2*a(A252373(k)), otherwise, a(n) = 1 + 2*a(n-A252373(n)-1).

cp's OEIS Frontend

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.

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A245701 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = 2a(n), a(A091242(n)) = 2a(n)+1, where A014580(n) = binary code for n-th irreducible polynomial over GF(2), A091242(n) = binary code for n-th reducible polynomial over GF(2).

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A252757 Permutation of natural numbers: a(1)=1, and for n>1, if n is k-th number whose largest prime factor is less than the square of its smallest prime factor [i.e., n = A251726(k)], a(n) = 2a(k), otherwise, when n = A251727(k), a(n) = 1 + 2a(k).

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cp's OEIS Frontend

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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A245701 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = 2*a(n), a(A091242(n)) = 2*a(n)+1, where A014580(n) = binary code for n-th irreducible polynomial over GF(2), A091242(n) = binary code for n-th reducible polynomial over GF(2).

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Author

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Programs

PARI

Scheme

Formula

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Author

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Formula

A252757 Permutation of natural numbers: a(1)=1, and for n>1, if n is k-th number whose largest prime factor is less than the square of its smallest prime factor [i.e., n = A251726(k)], a(n) = 2*a(k), otherwise, when n = A251727(k), a(n) = 1 + 2*a(k).

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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.

A245701 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = 2a(n), a(A091242(n)) = 2a(n)+1, where A014580(n) = binary code for n-th irreducible polynomial over GF(2), A091242(n) = binary code for n-th reducible polynomial over GF(2).

A252757 Permutation of natural numbers: a(1)=1, and for n>1, if n is k-th number whose largest prime factor is less than the square of its smallest prime factor [i.e., n = A251726(k)], a(n) = 2a(k), otherwise, when n = A251727(k), a(n) = 1 + 2a(k).