A075378 -id:A075378 - cp's OEIS frontend

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.

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].

A088610 Starting with n = 1, a(n) is the smallest squarefree number not included earlier if n is odd, else n is the smallest nonsquarefree number.

Original entry on oeis.org

1, 4, 2, 8, 3, 9, 5, 12, 6, 16, 7, 18, 10, 20, 11, 24, 13, 25, 14, 27, 15, 28, 17, 32, 19, 36, 21, 40, 22, 44, 23, 45, 26, 48, 29, 49, 30, 50, 31, 52, 33, 54, 34, 56, 35, 60, 37, 63, 38, 64, 39, 68, 41, 72, 42, 75, 43, 76, 46, 80, 47, 81, 51, 84, 53, 88, 55, 90, 57, 92, 58, 96

Offset: 1

Amarnath Murthy, Oct 16 2003

nonn

From Antti Karttunen, Jun 04 2014: (Start)
Squarefree (A005117) and nonsquarefree numbers (A013929) interleaved, the former at odd n and the latter at even n.
A243344 is a a "recursivized" variant of this permutation. Like this one, it also satisfies the given simple identity linking the parity of n with the Moebius mu-function. (End)

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers.

Inverse: A243352.
Bisections: A005117, A013929.
Similar permutations: A075378, A073846, A237056, A072061, A094077, A167151, A088609, A243344.
Cf. A000035, A008966.

With[{max = 100}, s = Select[Range[max], SquareFreeQ]; ns = Complement[Range[max], s]; Riffle[s[[1 ;; Length[ns]]], ns]] (* Amiram Eldar, Mar 04 2024 *)

(define (A088610 n) (if (even? n) (A013929 (/ n 2)) (A005117 (/ (+ 1 n) 2))))

From Antti Karttunen, Jun 04 2014: (Start)
a(2n) = A013929(n), a(2n-1) = A005117(n).
For all n, A008966(a(n)) = A000035(n), or equally, mu(a(n)) = n modulo 2, where mu is Moebius mu (A008683). (End)

More terms from Ray Chandler, Oct 18 2003

cp's OEIS Frontend

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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A088610 Starting with n = 1, a(n) is the smallest squarefree number not included earlier if n is odd, else n is the smallest nonsquarefree number.

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