A378357 Distance from n to the least non perfect power >= n.

1, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Nov 24 2024

Keywords

• nonn

Comments

Perfect powers (A001597) are 1 and numbers with a proper integer root, complement A007916.

All terms are <= 2 because the only adjacent perfect powers are 8 and 9.

Crossrefs

The version for prime numbers is A007920, subtraction of A159477 or A007918.

The version for perfect powers is A074984, subtraction of A377468.

The version for squarefree numbers is A081221, subtraction of A067535.

Subtracting from n gives A378358, opposite A378363.

The opposite version is A378364.

The version for nonsquarefree numbers is A378369, subtraction of A120327.

The version for prime powers is A378370, subtraction of A000015.

The version for non prime powers is A378371, subtraction of A378372.

The version for composite numbers is A378456, subtraction of A113646.

A000961 lists the powers of primes, differences A057820.

A001597 lists the perfect powers, differences A053289, seconds A376559.

A007916 lists the non perfect powers, differences A375706, seconds A376562.

A069623 counts perfect powers <= n.

A076411 counts perfect powers < n.

A377432 counts perfect powers between primes, zeros A377436.

Cf. A013597, A013603, A013632, A031218, A045542, A052410, A081676, A131605, A188951, A216765, A375702, A375736.

Programs

Mathematica

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[NestWhile[#+1&,n,#>1&&perpowQ[#]&]-n,{n,100}]

Python

from sympy import perfect_power
def A378357(n): return 0 if n>1 and perfect_power(n)==False else 1 if perfect_power(n+1)==False else 2 # Chai Wah Wu, Nov 27 2024

Formula

a(n) = n - A378358(n).