A319520 Starts of strictly increasing runs of 0's in Mertens's function A002321.

2, 39, 331, 422, 45371, 22898822, 871469945, 1319506393071, 26296710367071

Offset: 1

Author

Torlach Rush, Sep 21 2018

Keywords

• nonn
• more

Comments

a(n) is always squarefree.

It is not known whether this sequence is infinite. Sequence A045882 is infinite but it appears that increasing runs of consecutive nonsquarefree numbers thin out very quickly. The requirement that the runs consist of 0's is much stronger and makes it uncertain whether this sequence is also infinite.

Examples

			2 is a term because M(2) = 0.
39 is a term because M(39) = M(40) = 0.
331 is a term because M(331) = M(332) = M(333) = 0.
422 is a term because M(422) = ... = M(425) = 0.
45371 is a term because M(45371) = ... = M(45376) = 0.
		

Crossrefs

Cf. A002321, A045882.

Programs

Mathematica

With[{s = Map[Boole[# == 0] &, Accumulate@ Array[MoebiusMu, 10^5]]}, Union@ Array[SequencePosition[s, ConstantArray[1, #]][[1, 1]] &, 5]] (* Michael De Vlieger, Sep 26 2018 *)

PARI

M=S=R=0;for(n=1,oo,if(!M+=moebius(n),S||S=n,S,n-S>R&&print1(S",")+R=n-S;S=0)) \\ M. F. Hasler, Nov 23 2018

Extensions

a(6)-a(7) from Amiram Eldar, Sep 26 2018

a(8)-a(9) from Henri Lifchitz, Mar 09 2025