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Powers of squarefree semiprimes that are not squarefree.
3

%I #14 Aug 20 2024 01:59:31

%S 36,100,196,216,225,441,484,676,1000,1089,1156,1225,1296,1444,1521,

%T 2116,2601,2744,3025,3249,3364,3375,3844,4225,4761,5476,5929,6724,

%U 7225,7396,7569,7776,8281,8649,8836,9025,9261,10000,10648,11236,12321,13225,13924,14161,14884

%N Powers of squarefree semiprimes that are not squarefree.

%H Amiram Eldar, <a href="/A303661/b303661.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Squarefree.html">Squarefree</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Semiprime.html">Semiprime</a>.

%F Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A006881(n)-1)*A006881(n)) = Sum_{k>=2} (P(k)^2 - P(2*k))/2 = 0.07160601536406295068..., where P(k) is the prime zeta function. - _Amiram Eldar_, Feb 12 2021

%e 1089 is in the sequence because 1089 = 3^2*11^2.

%e 1296 is in the sequence because 1296 = 2^4*3^4.

%t Select[Range[15000], Length[Union[FactorInteger[#][[All, 2]]]] == 1 && PrimeNu[#] == 2 && ! SquareFreeQ[#] &]

%t seq[max_] := Module[{sp = Select[Range[Floor@Sqrt[max]], SquareFreeQ[#] && PrimeNu[#] == 2 &], s = {}}, Do[s = Join[s, sp[[k]]^Range[2, Floor@Log[sp[[k]], max]]], {k, 1, Length[sp]}]; Union@s]; seq[10000] (* _Amiram Eldar_, Feb 12 2021 *)

%o (Python)

%o from math import isqrt

%o from sympy import primepi, primerange, integer_nthroot

%o def A303661(n):

%o def g(x): return int(-(t:=primepi(s:=isqrt(x)))-(t*(t-1)>>1)+sum(primepi(x//k) for k in primerange(1, s+1)))

%o def f(x): return n-1+x-sum(g(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length()))

%o kmin, kmax = 1,2

%o while f(kmax) >= kmax:

%o kmax <<= 1

%o while True:

%o kmid = kmax+kmin>>1

%o if f(kmid) < kmid:

%o kmax = kmid

%o else:

%o kmin = kmid

%o if kmax-kmin <= 1:

%o break

%o return kmax # _Chai Wah Wu_, Aug 19 2024

%Y Cf. A006881, A013929, A072774, A072777, A085155, A123711, A200511, A246547, A303606.

%K nonn

%O 1,1

%A _Ilya Gutkovskiy_, Apr 28 2018