A375702 - OEIS

A375702

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

2, 3, 6, 8, 1, 4, 3, 12, 14, 16, 18, 20, 3, 2, 15, 24, 26, 19, 8, 17, 12, 32, 34, 18, 17, 38, 40, 42, 27, 16, 46, 48, 50, 52, 54, 56, 58, 60, 38, 23, 64, 66, 68, 70, 34, 37, 74, 76, 78, 80, 46, 35, 84, 86, 88, 22, 67, 70, 9, 11, 94, 96, 98, 100, 102, 39, 64

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OFFSET

1,1

COMMENTS

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

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10000

FORMULA

For n > 2 we have a(n) = A053289(n+1) - 1.

EXAMPLE

The list of all non-perfect-powers, split into runs, begins:

2 3

5 6 7

10 11 12 13 14 15

17 18 19 20 21 22 23 24

28 29 30 31

33 34 35

37 38 39 40 41 42 43 44 45 46 47 48

Row n has length a(n), first A375703, last A375704, sum A375705.

MATHEMATICA

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;

Length/@Split[Select[Range[100], radQ], #1+1==#2&]//Most

(* Alternative: *)

-1 + DeleteCases[Differences@ Join[{1}, Union@ Flatten[Table[k^m, {m, 2, Log2[#]}, {k, 2, Surd[#, m]}] ] ] &[4000], 1] (* Michael De Vlieger, Jan 31 2026 *)

CROSSREFS

For nonsquarefree numbers we have A053797, anti-runs A373409.

For squarefree numbers we have A120992, anti-runs A373127.

For nonprime numbers we have A176246, anti-runs A373403.

For prime-powers we have A373675, anti-runs A373576.

For non-prime-powers we have A373678, anti-runs A373679.

The anti-run version is A375736, sum A375737.

For runs of non-perfect-powers (A007916):

- length: this sequence.

- first: A375703

- last: A375704

- sum: A375705