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

1,2

Comments

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

Links

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

Formula

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

Example

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

Mathematica

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

Prog

(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
(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

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.

Sequence in context: A202241 A248156 A352780 * A331368 A106177 A069226

Adjacent sequences: A375703 A375704 A375705 * A375707 A375708 A375709

Keyword

nonn

Author

Gus Wiseman, Aug 31 2024

Extensions

More terms from Antti Karttunen, Jan 19 2025

Status

approved