A375706 First differences of non-perfect-powers.

1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2

Offset: 1

Author

Gus Wiseman, Aug 31 2024

Keywords

• nonn

Comments

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

Examples

			The 5th non-perfect-power is 7, and the 6th is 10, so a(5) = 3.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000

Crossrefs

For prime-powers (A000961) we have A057820.

For perfect powers (A001597) we have A053289.

For nonprime numbers (A002808) we have A073783.

For squarefree numbers (A005117) we have A076259.

First differences of A007916.

For nonsquarefree numbers (A013929) we have A078147.

For non-prime-powers (A024619) we have A375708.

Positions of 1s are A375740, complement A375714.

Runs of non-perfect-powers:

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

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

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

- sum: A375705

Cf. A000040, A046933, A052410, A303707, A305630, A305631, A323054, A323088, A323090, A375736.

Programs

Mathematica

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Differences[Select[Range[100],radQ]]

PARI

up_to = 112;
A375706list(up_to) = { my(v=vector(up_to), pk=2, k=2, i=0); while(i<#v, k++; if(!ispower(k), i++; v[i] = k-pk; pk = k)); (v); };
v375706 = A375706list(up_to);
A375706(n) = v375706[n]; \\ Antti Karttunen, Jan 19 2025

Python

from itertools import count
from sympy import mobius, integer_nthroot, perfect_power
def A375706(n):
    def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return next(i for i in count(m+1) if not perfect_power(i))-m # Chai Wah Wu, Sep 09 2024

Formula

a(n) = A007916(n+1) - A007916(n).

Extensions

More terms from Antti Karttunen, Jan 19 2025