A375704 Maximum of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

3, 7, 15, 24, 26, 31, 35, 48, 63, 80, 99, 120, 124, 127, 143, 168, 195, 215, 224, 242, 255, 288, 323, 342, 360, 399, 440, 483, 511, 528, 575, 624, 675, 728, 783, 840, 899, 960, 999, 1023, 1088, 1155, 1224, 1295, 1330, 1368, 1443, 1520, 1599, 1680, 1727, 1763

Offset: 1

Author

Gus Wiseman, Aug 29 2024

Keywords

• nonn

Comments

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

Also numbers k > 0 such that k is a perfect power (A001597) but k+1 is not.

Examples

			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
  26
  28  29  30  31
  33  34  35
  37  38  39  40  41  42  43  44  45  46  47  48
Row n begins with A375703(n), ends with a(n), adds up to A375705(n), and has length A375702(n).
		

Crossrefs

For nonprime numbers: A006093, min A055670, anti-runs A068780, min A005381.

For prime numbers we have A045344.

Inserting 8 after 7 gives A045542.

For nonsquarefree numbers we have A072284(n) + 1, anti-runs A068781.

For squarefree numbers we have A373415, anti-runs A007674.

For prime-powers we have A373674 (min A373673), anti-runs A006549 (A120430).

Non-prime-powers: A373677 (min A373676), anti-runs A255346 (min A373575).

The anti-run version is A375739.

A001597 lists perfect-powers, differences A053289.

A046933 counts composite numbers between primes.

A375736 gives lengths of anti-runs of non-prime-powers, sums A375737.

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

- length: A375702 = A053289(n+1) - 1

- first: A375703 (same as A216765 with 2 exceptions)

- last: A375704 (this) (same as A045542 with 8 removed)

- sum: A375705

Cf. A053797, A053806, A061398, A061399, A251092, A373408, A375708, 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]&]

Formula

For n > 2 we have a(n) = A045542(n+1).