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A376120 Refactorable numbers that are perfect powers.

%I A376120 #12 Oct 05 2024 03:38:30
%S A376120 1,8,9,36,128,225,441,625,1089,1521,2025,2601,3249,3600,4761,5625,
%T A376120 6561,7569,7776,8100,8649,10000,12321,15129,16641,19881,21952,22500,
%U A376120 25281,26244,28224,31329,32400,32768,33489,35721,40401,45369,47961,50625,56169,62001,64000,71289,84681,90000
%N A376120 Refactorable numbers that are perfect powers.
%C A376120 Intersection of A001597 and A033950.
%e A376120 8 is a perfect power, as 8=2^3, and it is also a refactorable numbers, being divisible by its number of divisors (4).
%t A376120 Join[{1}, Select[Range[10^5], Divisible[#, DivisorSigma[0,#]]&&GCD@@FactorInteger[#][[All, 2]]>1&]]
%o A376120 (PARI) ok(n) = n==1 || (n%numdiv(n)==0&&ispower(n))
%o A376120 (Python)
%o A376120 from itertools import count, islice
%o A376120 from math import prod
%o A376120 from sympy import mobius, integer_nthroot, factorint
%o A376120 def A376120_gen(): # generator of terms
%o A376120     def bisection(f,kmin=0,kmax=1):
%o A376120         while f(kmax) > kmax: kmax <<= 1
%o A376120         while kmax-kmin > 1:
%o A376120             kmid = kmax+kmin>>1
%o A376120             if f(kmid) <= kmid:
%o A376120                 kmax = kmid
%o A376120             else:
%o A376120                 kmin = kmid
%o A376120         return kmax
%o A376120     def f(x): return int(x-1+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
%o A376120     m = 1
%o A376120     for n in count(1):
%o A376120         m = bisection(lambda x:f(x)+n,m,m)
%o A376120         if not m%prod(e+1 for e in factorint(m).values()): yield m
%o A376120 A376120_list = list(islice(A376120_gen(),40)) # _Chai Wah Wu_, Oct 04 2024
%Y A376120 Cf. A001597 (perfect powers), A033950 (refactorable numbers).
%K A376120 nonn
%O A376120 1,2
%A A376120 _Waldemar Puszkarz_, Sep 11 2024