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Squarefree kernels of powers of squarefree numbers.
4

%I #12 Aug 19 2024 02:20:49

%S 1,2,3,2,5,6,7,2,3,10,11,13,14,15,2,17,19,21,22,23,5,26,3,29,30,31,2,

%T 33,34,35,6,37,38,39,41,42,43,46,47,7,51,53,55,57,58,59,61,62,2,65,66,

%U 67,69,70,71,73,74,77,78,79,3,82,83,85,86,87,89,91,93,94,95,97,10,101

%N Squarefree kernels of powers of squarefree numbers.

%C a(n) = A007947(A072774(n));

%C A072774(n) = a(n)^A072776(n);

%C A072774(n) is squarefree iff A072774(n)=a(n).

%H Reinhard Zumkeller, <a href="/A072775/b072775.txt">Table of n, a(n) for n = 1..10000</a>

%o (Haskell)

%o a072775 n = a072775_list !! (n-1) -- a072775_list defined in A072774.

%o -- _Reinhard Zumkeller_, Apr 06 2014

%o (Python)

%o from math import isqrt, prod

%o from sympy import mobius, integer_nthroot, primefactors

%o def A072775(n):

%o def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))-1

%o def f(x): return n-2+x-sum(g(integer_nthroot(x,k)[0]) for k in range(1,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 prod(primefactors(kmax)) # _Chai Wah Wu_, Aug 19 2024

%Y Cf. A052410.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Jul 10 2002