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.

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

1,2

Comments

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)

Links

Antti Karttunen, Table of n, a(n) for n = 1..10001

Index entries for sequences that are permutations of the natural numbers

Formula

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)

Prog

(define (A088609 n) (cond ((<= n 1) n) ((zero? (A008966 n)) (A005117 (+ 1 (A057627 n)))) (else (A013929 (A013928 n))))) ;; Antti Karttunen, Jun 04 2014

Crossrefs

Cf. A026239, A088610-A243352, A243347.

Sequence in context: A020848 A133058 A011516 * A163813 A109815 A339571

Adjacent sequences: A088606 A088607 A088608 * A088610 A088611 A088612

Keyword

nonn

Author

Amarnath Murthy, Oct 16 2003

Extensions

More terms from Ray Chandler, Oct 18 2003

Status

approved