A088610 -id:A088610 - cp's OEIS frontend

A075379 Erroneous version of A088610.

2, 1, 3, 4, 5, 8, 6, 9, 7, 12, 10, 16, 11, 18, 13, 20, 14, 24, 15, 25, 17, 27, 19, 28, 21, 32, 22, 36, 23, 40

Offset: 1

dead

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

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.]

A243352 If n is k-th squarefree number [i.e., n = A005117(k)], a(n) = 2k-1; otherwise, when n is k-th nonsquarefree number [i.e., n = A013929(k)], a(n) = 2k.

Original entry on oeis.org

1, 3, 5, 2, 7, 9, 11, 4, 6, 13, 15, 8, 17, 19, 21, 10, 23, 12, 25, 14, 27, 29, 31, 16, 18, 33, 20, 22, 35, 37, 39, 24, 41, 43, 45, 26, 47, 49, 51, 28, 53, 55, 57, 30, 32, 59, 61, 34, 36, 38, 63, 40, 65, 42, 67, 44, 69, 71, 73, 46, 75, 77, 48, 50, 79, 81, 83, 52, 85, 87, 89

Offset: 1

Antti Karttunen, Jun 04 2014

Odd numbers occur (in order) at the positions given by squarefree numbers, A005117, and even numbers occur (in order) at the positions given by their complement, nonsquarefree numbers, A013929.

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

Inverse: A088610. Cf. A243343, A072062.

(define (A243352 n) (if (zero? (A008966 n)) (* 2 (A057627 n)) (+ (* 2 (A013928 n)) 1)))

If mu(n) = 0, a(n) = 2*A057627(n), otherwise, a(n) = 1 + 2 * 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, A000035(a(n)) = A008966(n) = A008683(n)^2, or equally, a(n) = mu(n) modulo 2.

A088609 a(1) = 1, a(n) is the smallest squarefree number not included earlier if n is not squarefree, else n is the smallest nonsquarefree number.

Original entry on oeis.org

1, 4, 8, 2, 9, 12, 16, 3, 5, 18, 20, 6, 24, 25, 27, 7, 28, 10, 32, 11, 36, 40, 44, 13, 14, 45, 15, 17, 48, 49, 50, 19, 52, 54, 56, 21, 60, 63, 64, 22, 68, 72, 75, 23, 26, 76, 80, 29, 30, 31, 81, 33, 84, 34, 88, 35, 90, 92, 96, 37, 98, 99, 38, 39, 100, 104, 108, 41, 112, 116

Offset: 1

Amarnath Murthy, Oct 16 2003

nonn

From Antti Karttunen, Jun 04 2014: (Start)
This is a self-inverse permutation (involution) of natural numbers.
After 1, nonsquarefree numbers occur (in monotonic order) at the positions given by squarefree numbers, A005117, and squarefree numbers occur (in monotonic order) at the positions given by their complement, nonsquarefree numbers, A013929.
(End)

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

Cf. A026239, A088610-A243352, A243347.

From Antti Karttunen, Jun 04 2014: (Start)
a(1), and for n>1, if mu(n) = 0, a(n) = A005117(1+A057627(n)), otherwise, a(n) =  A013929(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. [A property shared with A243347].
(End)

More terms from Ray Chandler, Oct 18 2003

cp's OEIS Frontend

A075379 Erroneous version of A088610.

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

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A243352 If n is k-th squarefree number [i.e., n = A005117(k)], a(n) = 2k-1; otherwise, when n is k-th nonsquarefree number [i.e., n = A013929(k)], a(n) = 2k.

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Formula

A088609 a(1) = 1, a(n) is the smallest squarefree number not included earlier if n is not squarefree, else n is the smallest nonsquarefree number.

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