A276108 - OEIS

%I #47 Nov 26 2024 18:11:11

%S 1,65536,43046721,68719476736,152587890625,2821109907456,

%T 33232930569601,281474976710656,10000000000000000,45949729863572161,

%U 150094635296999121,184884258895036416,665416609183179841,2177953337809371136,6568408355712890625,18446744073709551616

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

%C Old title was "Values of A117453(n) such that A175066(n) is not a prime number."

%C Terms are 1, 2^16, 3^16, 2^36, ...

%C 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 Chai Wah Wu, <a href="/A276108/b276108.txt">Table of n, a(n) for n = 1..10000</a>

%e 65536 = 2^16 is a term because there are 4 corresponding ways that are 2^16, 4^8, 16^4, 256^2.

%o (Python)

%o from sympy import mobius, integer_nthroot, isprime, divisor_count

%o def A276108(n):

%o     def bisection(f,kmin=0,kmax=1):

%o         while f(kmax) > kmax: kmax <<= 1

%o         while kmax-kmin > 1:

%o             kmid = kmax+kmin>>1

%o             if f(kmid) <= kmid:

%o                 kmax = kmid

%o             else:

%o                 kmin = kmid

%o         return kmax

%o     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     return bisection(f,n,n) # _Chai Wah Wu_, Nov 25 2024

%Y Cf. A117453, A154893, A175066.

%K nonn,changed

%O 1,2

%A _Altug Alkan_, Aug 27 2016

%E New title from _Charlie Neder_, Mar 04 2019

%E a(5)-a(16) from _Chai Wah Wu_, Nov 25 2024