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A182853 Squarefree composite integers and powers of squarefree composite integers.

%I A182853 #29 Feb 16 2025 08:33:13
%S A182853 6,10,14,15,21,22,26,30,33,34,35,36,38,39,42,46,51,55,57,58,62,65,66,
%T A182853 69,70,74,77,78,82,85,86,87,91,93,94,95,100,102,105,106,110,111,114,
%U A182853 115,118,119,122,123,129,130,133,134,138,141,142,143,145,146,154,155,158,159,161
%N A182853 Squarefree composite integers and powers of squarefree composite integers.
%C A182853 Numbers that require exactly three iterations to reach a fixed point under the x -> A181819(x) map. In each case, 2 is the fixed point that is reached. (1 is the other fixed point of the x -> A181819(x) map.) Cf. A182850.
%C A182853 Numbers such that A001221(n) > 1 and A071625(n) = 1.
%H A182853 Antti Karttunen, <a href="/A182853/b182853.txt">Table of n, a(n) for n = 1..10000</a>
%H A182853 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FixedPoint.html">Fixed Point</a>
%H A182853 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Map.html">Map</a>
%o A182853 (Scheme)
%o A182853 (define A182853 (MATCHING-POS 1 1 (lambda (n) (= 3 (A182850 n))))) ;; After the alternative definition of the sequence given by the original author. Requires also MATCHING-POS macro from my IntSeq-library - _Antti Karttunen_, Feb 05 2016
%o A182853 (PARI) isoka(n) = (omega(n) > 1) && issquarefree(n); \\ A120944
%o A182853 isok(n) = isoka(n) || (ispower(n,,&k) && isoka(k)); \\ _Michel Marcus_, Jun 24 2017
%o A182853 (Python)
%o A182853 from math import isqrt
%o A182853 from sympy import mobius, primepi, integer_nthroot
%o A182853 def A182853(n):
%o A182853     def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))-primepi(x))
%o A182853     def f(x): return n-2+x+(y:=x.bit_length())-sum(g(integer_nthroot(x,k)[0]) for k in range(1,y))
%o A182853     kmin, kmax = 1,2
%o A182853     while f(kmax) >= kmax:
%o A182853         kmax <<= 1
%o A182853     while True:
%o A182853         kmid = kmax+kmin>>1
%o A182853         if f(kmid) < kmid:
%o A182853             kmax = kmid
%o A182853         else:
%o A182853             kmin = kmid
%o A182853         if kmax-kmin <= 1:
%o A182853             break
%o A182853     return kmax # _Chai Wah Wu_, Aug 19 2024
%Y A182853 Numbers n such that A182850(n) = 3. See also A182854, A182855.
%Y A182853 Subsequence of A072774 and A182851.
%Y A182853 Cf. A120944.
%K A182853 nonn,easy
%O A182853 1,1
%A A182853 _Matthew Vandermast_, Jan 04 2011