cp's OEIS Frontend

A303606 Powers of composite squarefree numbers that are not squarefree.

%I A303606 #15 Feb 16 2025 08:33:54
%S A303606 36,100,196,216,225,441,484,676,900,1000,1089,1156,1225,1296,1444,
%T A303606 1521,1764,2116,2601,2744,3025,3249,3364,3375,3844,4225,4356,4761,
%U A303606 4900,5476,5929,6084,6724,7225,7396,7569,7776,8281,8649,8836,9025,9261,10000,10404,10648,11025,11236
%N A303606 Powers of composite squarefree numbers that are not squarefree.
%H A303606 Amiram Eldar, <a href="/A303606/b303606.txt">Table of n, a(n) for n = 1..10000</a>
%H A303606 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimePower.html">Prime Power</a>.
%H A303606 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree</a>.
%F A303606 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
%e A303606 196 is in the sequence because 196 = 2^2*7^2.
%e A303606 4900 is in the sequence because 4900 = 2^2*5^2*7^2.
%t A303606 Select[Range[12000], Length[Union[FactorInteger[#][[All, 2]]]] == 1 && ! SquareFreeQ[#] && ! PrimePowerQ[#] &]
%t A303606 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 *)
%o A303606 (Python)
%o A303606 from math import isqrt
%o A303606 from sympy import mobius, primepi, integer_nthroot
%o A303606 def A303606(n):
%o A303606     def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))-primepi(x))
%o A303606     def f(x): return n-3+x+(y:=x.bit_length())-sum(g(integer_nthroot(x,k)[0]) for k in range(2,y))
%o A303606     kmin, kmax = 1,2
%o A303606     while f(kmax) >= kmax:
%o A303606         kmax <<= 1
%o A303606     while True:
%o A303606         kmid = kmax+kmin>>1
%o A303606         if f(kmid) < kmid:
%o A303606             kmax = kmid
%o A303606         else:
%o A303606             kmin = kmid
%o A303606         if kmax-kmin <= 1:
%o A303606             break
%o A303606     return kmax # _Chai Wah Wu_, Aug 19 2024
%Y A303606 Intersection of A024619 and A072777.
%Y A303606 Intersection of A072774 and A126706.
%Y A303606 Intersection of A013929 and A182853.
%Y A303606 Cf. A000469, A001597, A005117, A120944, A286708.
%K A303606 nonn
%O A303606 1,1
%A A303606 _Ilya Gutkovskiy_, Apr 26 2018