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A303606
Powers of composite squarefree numbers that are not squarefree.
11
36, 100, 196, 216, 225, 441, 484, 676, 900, 1000, 1089, 1156, 1225, 1296, 1444, 1521, 1764, 2116, 2601, 2744, 3025, 3249, 3364, 3375, 3844, 4225, 4356, 4761, 4900, 5476, 5929, 6084, 6724, 7225, 7396, 7569, 7776, 8281, 8649, 8836, 9025, 9261, 10000, 10404, 10648, 11025, 11236
OFFSET
1,1
Eric Weisstein's World of Mathematics, Prime Power.
Eric Weisstein's World of Mathematics, Squarefree.
FORMULA
Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A120944(n)-1)*A120944(n)) = Sum_{k>=2} (zeta(k)/zeta(2*k) - P(k) - 1) = 0.07547719891508850482..., where P(k) is the prime zeta function. - Amiram Eldar, Feb 12 2021
EXAMPLE
196 is in the sequence because 196 = 2^2*7^2.
4900 is in the sequence because 4900 = 2^2*5^2*7^2.
MATHEMATICA
Select[Range[12000], Length[Union[FactorInteger[#][[All, 2]]]] == 1 && ! SquareFreeQ[#] && ! PrimePowerQ[#] &]
seq[max_] := Module[{sp = Select[Range[Floor@Sqrt[max]], SquareFreeQ[#] && PrimeNu[#] > 1 &], s = {}}, Do[s = Join[s, sp[[k]]^Range[2, Floor@Log[sp[[k]], max]]], {k, 1, Length[sp]}]; Union@s]; seq[10^4] (* Amiram Eldar, Feb 12 2021 *)
PROG
(Python)
from math import isqrt
from sympy import mobius, primepi, integer_nthroot
def A303606(n):
def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))-primepi(x))
def f(x): return n-3+x+(y:=x.bit_length())-sum(g(integer_nthroot(x, k)[0]) for k in range(2, y))
kmin, kmax = 1, 2
while f(kmax) >= kmax:
kmax <<= 1
while True:
kmid = kmax+kmin>>1
if f(kmid) < kmid:
kmax = kmid
else:
kmin = kmid
if kmax-kmin <= 1:
break
return kmax # Chai Wah Wu, Aug 19 2024
CROSSREFS
Intersection of A024619 and A072777.
Intersection of A072774 and A126706.
Intersection of A013929 and A182853.
Sequence in context: A044604 A375144 A340017 * A303661 A175391 A030627
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 26 2018
STATUS
approved

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Last modified September 20 00:39 EDT 2024. Contains 376015 sequences. (Running on oeis4.)