A288118 Lexicographically earliest sequence of distinct positive terms such that, for any i and j > 0, a(i*j) != a(i) * a(j).

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

Offset: 1

Author

Rémy Sigrist, Jun 05 2017

Keywords

• nonn

Comments

If we drop the unicity constraint, then we obtain the Liouville's function (A007421).

This sequence is a permutation of the natural numbers (we can always choose the least value not yet seen at prime positions).

Conjecturally:

- the sequence is self-inverse,

- | a(n) - n | <= 1 for any n > 0,

- | a(i)*a(j) - i*j | <> 1 for any i > 0 and j > 0,

- a(n) = n+1 iff a(n+1) = n.

a(6) = a(1) * a(2) * a(3).

This sequence has connections with A288119; here we avoid a(i)*a(j) = a(i*j), there a(i)+a(j) = a(i*j).

Examples

			a(1) cannot equal 1 as a(1*1) != a(1)*a(1); a(1) = 2 is acceptable.
a(2) cannot equal a(1); a(2) = 1 is acceptable.
a(3) cannot equal a(1), a(2); a(3) = 3 is acceptable.
a(4) cannot equal a(1)...a(3), a(2)^2; a(4) = 5 is acceptable.
a(5) cannot equal a(1)...a(4); a(5) = 4 is acceptable.
a(6) cannot equal a(1)...a(5), a(2)*a(3); a(6) = 6 is acceptable.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007421, A288119.