A375736 Length of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Gus Wiseman, Sep 10 2024

Keywords

• nonn

Comments

Non-perfect-powers (A007916) are numbers with no proper integer roots.

An anti-run of a sequence is an interval of positions at which consecutive terms differ by more than one.

Examples

			The initial anti-runs are the following, whose lengths are a(n):
  (2)
  (3,5)
  (6)
  (7,10)
  (11)
  (12)
  (13)
  (14)
  (15,17)
  (18)
  (19)
  (20)
  (21)
  (22)
  (23)
  (24,26,28)
		

Crossrefs

For squarefree numbers we have A373127, runs A120992.

For nonprime numbers we have A373403, runs A176246.

For nonsquarefree numbers we have A373409, runs A053797.

For prime-powers we have A373576, runs A373675.

For non-prime-powers (exclusive) we have A373672, runs A110969.

For runs instead of anti-runs we have A375702.

For anti-runs of non-perfect-powers:

- length: A375736 (this)

- first: A375738

- last: A375739

- sum: A375737

For runs of non-perfect-powers:

- length: A375702

- first: A375703

- last: A375704

- sum: A375705

A001597 lists perfect-powers, differences A053289.

A007916 lists non-perfect-powers, differences A375706.

Cf. A007674, A045542, A046933, A216765, A251092, A373679, A375714.

Programs

Mathematica

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Length/@Split[Select[Range[100],radQ],#1+1!=#2&]//Most