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A276108 Numbers expressible as perfect powers in a composite number of ways.

%I A276108 #47 Nov 26 2024 18:11:11
%S A276108 1,65536,43046721,68719476736,152587890625,2821109907456,
%T A276108 33232930569601,281474976710656,10000000000000000,45949729863572161,
%U A276108 150094635296999121,184884258895036416,665416609183179841,2177953337809371136,6568408355712890625,18446744073709551616
%N A276108 Numbers expressible as perfect powers in a composite number of ways.
%C A276108 Old title was "Values of A117453(n) such that A175066(n) is not a prime number."
%C A276108 Terms are 1, 2^16, 3^16, 2^36, ...
%C A276108 Numbers m^k, where m is not a perfect power and k is a composite number in A154893 or 0. - _Charlie Neder_, Mar 02 2019
%H A276108 Chai Wah Wu, <a href="/A276108/b276108.txt">Table of n, a(n) for n = 1..10000</a>
%e A276108 65536 = 2^16 is a term because there are 4 corresponding ways that are 2^16, 4^8, 16^4, 256^2.
%o A276108 (Python)
%o A276108 from sympy import mobius, integer_nthroot, isprime, divisor_count
%o A276108 def A276108(n):
%o A276108     def bisection(f,kmin=0,kmax=1):
%o A276108         while f(kmax) > kmax: kmax <<= 1
%o A276108         while kmax-kmin > 1:
%o A276108             kmid = kmax+kmin>>1
%o A276108             if f(kmid) <= kmid:
%o A276108                 kmax = kmid
%o A276108             else:
%o A276108                 kmin = kmid
%o A276108         return kmax
%o A276108     def f(x): return int(n+sum(mobius(k)*(integer_nthroot(x,k)[0]-1+sum(integer_nthroot(x,i*k)[0]-1 for i in range(2,(x//k).bit_length()) if isprime(i) or isprime(divisor_count(i)-1))) for k in range(1,x.bit_length())))
%o A276108     return bisection(f,n,n) # _Chai Wah Wu_, Nov 25 2024
%Y A276108 Cf. A117453, A154893, A175066.
%K A276108 nonn
%O A276108 1,2
%A A276108 _Altug Alkan_, Aug 27 2016
%E A276108 New title from _Charlie Neder_, Mar 04 2019
%E A276108 a(5)-a(16) from _Chai Wah Wu_, Nov 25 2024