A375739 Maximum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

2, 5, 6, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 23, 28, 29, 30, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88

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.

Also non-perfect-powers x such that x + 1 is also a non-perfect-power.

Examples

			The initial anti-runs are the following, whose maxima 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 nonprime numbers we have A068780, runs A006093 with 2 removed.

For squarefree numbers we have A007674, runs A373415.

For nonsquarefree numbers we have A068781, runs A072284 minus 1 and shifted.

For prime-powers we have A006549, runs A373674.

For non-prime-powers we have A255346, runs A373677.

For anti-runs of non-perfect-powers:

- length: A375736

- first: A375738

- last: A375739 (this)

- 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. A045542, A046933, A053797, A216765, A373576, A373679, A375714.

Programs

Mathematica

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