A350033 a(n) is the smallest positive integer k not occurring earlier such that Möbius(k) == n (mod 3).

1, 2, 4, 6, 3, 8, 10, 5, 9, 14, 7, 12, 15, 11, 16, 21, 13, 18, 22, 17, 20, 26, 19, 24, 33, 23, 25, 34, 29, 27, 35, 30, 28, 38, 31, 32, 39, 37, 36, 46, 41, 40, 51, 42, 44, 55, 43, 45, 57, 47, 48, 58, 53, 49, 62, 59, 50, 65, 61, 52, 69, 66, 54, 74, 67, 56, 77, 70

Offset

1,2

Comments

Permutation of the positive integers.

In other words, a(n) is the least positive unused integer such that Möbius(a(n)) is respectively 1, -1, 0, 1, -1, 0, 1, -1, 0, ... for n > 0.

Links

Winston de Greef, Table of n, a(n) for n = 1..10000

Example

a(1) = 1 because 1 is the smallest positive integer with Möbius(1) = 1.
For a(2) we search for the smallest positive integer k not already in the sequence with Möbius(k) = -1. So, a(2) = 2.
Next, we search for Möbius(k) = 0 and a(3) = 6. We continue by asking for the smallest k such that Möbius(k) = 1, -1, 0, 1, -1, 0, ... and so on.

Mathematica

a[1]=1; a[n_]:=a[n]=(k=1; While[MemberQ[Array[a, n-1], k]||Mod[n, 3, -1]!=MoebiusMu@k, k++]; k); Array[a, 100]

Prog

(PARI) lista(n)=my(v=vector(n), k=[1, 1, 1]); for(i=1, n, my(m=i%3); while((moebius(k[m+1])-m) % 3, k[m+1]++); v[i]=k[m+1]; k[m+1]++; ); v \\ Winston de Greef, Mar 25 2023

Crossrefs

Cf. A008683.

Sequence in context: A101543 A073900 A361629 * A351625 A352976 A363956

Adjacent sequences: A350030 A350031 A350032 * A350034 A350035 A350036

Keyword

nonn

Author

Giorgos Kalogeropoulos, Dec 09 2021

Status

approved