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A126706 Positive integers which are neither squarefree integers nor prime powers.

%I A126706 #28 Aug 15 2024 02:02:21
%S A126706 12,18,20,24,28,36,40,44,45,48,50,52,54,56,60,63,68,72,75,76,80,84,88,
%T A126706 90,92,96,98,99,100,104,108,112,116,117,120,124,126,132,135,136,140,
%U A126706 144,147,148,150,152,153,156,160,162,164,168,171,172,175,176,180,184,188
%N A126706 Positive integers which are neither squarefree integers nor prime powers.
%H A126706 Michael De Vlieger, <a href="/A126706/b126706.txt">Table of n, a(n) for n = 1..10000</a>
%e A126706 45 is in the sequence because 45=3^2*5, i.e., neither squarefree nor a prime power.
%p A126706 with(numtheory): a:=proc(n) if mobius(n)=0 and nops(factorset(n))>1 then n else fi end: seq(a(n), n=1..230); # _Emeric Deutsch_, Feb 17 2007
%t A126706 Select[Range[200], Max @@ Last /@ FactorInteger[ # ] >1 && Length[FactorInteger[ # ]] > 1 &] (* _Ray Chandler_, Feb 17 2007 *)
%t A126706 Select[Range[200],!SquareFreeQ[#]&&!PrimePowerQ[#]&] (* _Harvey P. Dale_, Aug 05 2023 *)
%o A126706 (PARI) isok(k) = !issquarefree(k) && !isprimepower(k); \\ _Michel Marcus_, Nov 02 2022
%o A126706 (Python)
%o A126706 from math import isqrt
%o A126706 from sympy import primepi, integer_nthroot, mobius
%o A126706 def A126706(n):
%o A126706     def f(x): return int(n+sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length()))+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
%o A126706     m, k = n, f(n)
%o A126706     while m != k:
%o A126706         m, k = k, f(k)
%o A126706     return m # _Chai Wah Wu_, Aug 15 2024
%Y A126706 Cf. A005117, A000961, A059404.
%K A126706 nonn
%O A126706 1,1
%A A126706 _Leroy Quet_, Feb 11 2007
%E A126706 Extended by _Emeric Deutsch_ and _Ray Chandler_, Feb 17 2007