A378357 - OEIS

%I #14 Nov 27 2024 18:57:59

%S 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,

%T 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,

%U 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0

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

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

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

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

%t perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;

%t Table[NestWhile[#+1&,n,#>1&&perpowQ[#]&]-n,{n,100}]

%o (Python)

%o from sympy import perfect_power

%o 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

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

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

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

%Y Subtracting from n gives A378358, opposite A378363.

%Y The opposite version is A378364.

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

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

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

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

%Y A000961 lists the powers of primes, differences A057820.

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

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

%Y A069623 counts perfect powers <= n.

%Y A076411 counts perfect powers < n.

%Y A377432 counts perfect powers between primes, zeros A377436.

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

%K nonn

%O 1,8

%A _Gus Wiseman_, Nov 24 2024