A375703 Minimum of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

2, 5, 10, 17, 26, 28, 33, 37, 50, 65, 82, 101, 122, 126, 129, 145, 170, 197, 217, 226, 244, 257, 290, 325, 344, 362, 401, 442, 485, 513, 530, 577, 626, 677, 730, 785, 842, 901, 962, 1001, 1025, 1090, 1157, 1226, 1297, 1332, 1370, 1445, 1522, 1601, 1682, 1729

Offset: 1

Author

Gus Wiseman, Aug 28 2024

Keywords

• nonn

Comments

Non-perfect-powers A007916 are numbers without a proper integer root.

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 has length A375702, first a(n), last A375704, sum A375705.
		

Crossrefs

For prime numbers we have A045344.

For nonsquarefree numbers we have A053806, anti-runs A373410.

For nonprime numbers we have A055670, anti-runs A005381.

For squarefree numbers we have A072284, anti-runs A373408.

The anti-run version is A216765 (same as A375703 with 2 exceptions).

For non-prime-powers we have A373673, anti-runs A120430.

For prime-powers we have A373676, anti-runs A373575.

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

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

- first: A375703 (this)

- last: A375704

- sum: A375705

A001597 lists perfect-powers, differences A053289.

A007916 lists non-perfect-powers, differences A375706.

A046933 counts composite numbers between primes.

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

Cf. A007674, A045542, A375708, A375713, A375714.

Programs

Mathematica

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

Numbers k > 0 such that k-1 is a perfect power (A001597) but k is not.