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A076400 Number of divisors of n-th perfect power.

%I A076400 #13 Feb 16 2025 08:32:47
%S A076400 1,3,4,3,5,3,4,6,9,3,7,5,9,3,4,8,15,3,9,16,9,6,9,3,15,4,3,15,9,9,10,3,
%T A076400 21,5,9,7,15,3,27,3,16,11,9,9,9,25,4,3,9,9,21,3,28,27,3,15,15,12,9,8,
%U A076400 4,3,27,5,15,9,15,16,3,21,9,6,21,9,9,16,3,45,3,9,15,13,9,27,3,15,9,27,4
%N A076400 Number of divisors of n-th perfect power.
%H A076400 Michael De Vlieger, <a href="/A076400/b076400.txt">Table of n, a(n) for n = 1..10000</a>
%H A076400 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PerfectPower.html">Perfect Powers</a>.
%H A076400 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>.
%F A076400 a(n) = A000005(A001597(n)).
%t A076400 DivisorSigma[0, {1}~Join~Select[Range[5000], GCD @@ FactorInteger[#][[All, -1]] > 1 &]] (* _Michael De Vlieger_, Dec 16 2021 *)
%o A076400 (Python)
%o A076400 from sympy import mobius, integer_nthroot, divisor_count
%o A076400 def A076400(n):
%o A076400     def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
%o A076400     kmin, kmax = 1,2
%o A076400     while f(kmax) >= kmax:
%o A076400         kmax <<= 1
%o A076400     while True:
%o A076400         kmid = kmax+kmin>>1
%o A076400         if f(kmid) < kmid:
%o A076400             kmax = kmid
%o A076400         else:
%o A076400             kmin = kmid
%o A076400         if kmax-kmin <= 1:
%o A076400             break
%o A076400     return int(divisor_count(kmax)) # _Chai Wah Wu_, Aug 14 2024
%Y A076400 Cf. A000005, A001597, A076398, A076399, A076401.
%K A076400 nonn
%O A076400 1,2
%A A076400 _Reinhard Zumkeller_, Oct 09 2002