A378363 Greatest number <= n that is 1 or not a perfect-power.

1, 2, 3, 3, 5, 6, 7, 7, 7, 10, 11, 12, 13, 14, 15, 15, 17, 18, 19, 20, 21, 22, 23, 24, 24, 26, 26, 28, 29, 30, 31, 31, 33, 34, 35, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 63, 65, 66, 67

Offset: 1

Author

Gus Wiseman, Nov 24 2024

Keywords

• nonn
• changed

Comments

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

Examples

			In the non-perfect-powers ... 5, 6, 7, 10, 11 ... the greatest term <= 8 is 7, so a(8) = 7.
		

Crossrefs

The union is A007916, complement A001597.

The version for prime numbers is A007917 or A151799, opposite A159477.

The version for prime-powers is A031218, opposite A000015.

The version for squarefree numbers is A067535, opposite A070321.

The version for perfect-powers is A081676, opposite A377468.

The version for composite numbers is A179278, opposite A113646.

Terms appearing multiple times are A375704, opposite A375703.

The run-lengths are A375706.

Terms appearing only once are A375739, opposite A375738.

The version for nonsquarefree numbers is A378033, opposite A120327.

The opposite version is A378358.

Subtracting n gives A378364, opposite A378357.

The version for non-prime-powers is A378367 (subtracted A378371), opposite A378372.

A000040 lists the primes, differences A001223.

A000961 lists the powers of primes, differences A057820.

A001597 lists the perfect-powers, differences A053289.

A007916 lists the non-perfect-powers, differences A375706.

A069623 counts perfect-powers <= n.

A076411 counts perfect-powers < n.

A131605 lists perfect-powers that are not prime-powers.

A377432 counts perfect-powers between primes, zeros A377436.

Cf. A007918, A014210, A014234, A045542, A052410, A065514, A076412, A162966, A188951, A345531.

Programs

Mathematica

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

Python

from sympy import mobius, integer_nthroot
def A378363(n):
    def f(x): return int(1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    a = n-f(n)
    m, k = a, f(a)+a
    while m != k: m, k = k, f(k)+a
    return m # Chai Wah Wu, Nov 26 2024