A303946 - OEIS

A303946

Numbers that are neither squarefree nor perfect powers.

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

First differs from A059404 at a(40) = 147, A059404(40) = 144.

First differs from A126706 at a(6) = 40, A126706(6) = 36.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

FORMULA

a(n) ~ n/k, where k = 1 - 1/zeta(2) = 1 - 6/Pi^2 = A229099. - Charles R Greathouse IV, Jun 01 2018

MAPLE

filter:= proc(n) local F;

F:= map(t->t[2], ifactors(n)[2]);

max(F)>1 and igcd(op(F))=1

end proc:

select(filter, [$1..1000]); # Robert Israel, May 06 2018

MATHEMATICA

Select[Range[200], !SquareFreeQ[#] && GCD@@FactorInteger[#][[All, 2]] == 1 &]

PROG

(PARI) isok(n) = !issquarefree(n) && !ispower(n); \\ Michel Marcus, May 05 2018

(Python)

from math import isqrt

from sympy import mobius, integer_nthroot

def A303946(n):

def f(x): return int(n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+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 m # Chai Wah Wu, Aug 19 2024

CROSSREFS

Cf. A000009, A000837, A001597, A005117, A007916, A013929, A047966, A052409, A052410, A072774, A073576, A126706, A132350.

Sequence in context: A323055 A059404 A376250 * A360246 A242416 A360248

Adjacent sequences: A303943 A303944 A303945 * A303947 A303948 A303949

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 03 2018

STATUS

approved