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A175066 a(1) = 1, for n >= 2: a(n) = number of ways h to write perfect powers A117453(n) as m^k (m >= 2, k >= 2).

%I A175066 #30 Nov 25 2024 09:08:07
%S A175066 1,2,3,2,3,2,2,3,3,2,2,5,3,2,2,3,3,2,2,2,3,2,3,2,3,4,2,2,3,2,2,2,2,5,
%T A175066 2,2,3,2,5,2,2,2,2,3,5,2,2,2,3,3,2,2,2,3,2,2,2,3,2,2,2,2,3,2,3,3,2,2,
%U A175066 3,2,2,2,3,2,2,2,3,2,2,2,2,3,2,2,2,3,2,2,7,2,2,2,2,2,2,3,2,2,2,2,2,2,2,3,2
%N A175066 a(1) = 1, for n >= 2: a(n) = number of ways h to write perfect powers A117453(n) as m^k (m >= 2, k >= 2).
%C A175066 Perfect powers with first occurrence of h >= 2: 16, 64, 65536, 4096, ... (A175065)
%C A175066 a(n) for n>1 is the subsequence of A253642 formed by the terms which exceed 1; equivalently, a(n)+1 for n>1 is the subsequence of A175064 formed by the terms which exceed 2. Also, sum of a(n)-1 over such n that A117453(n)<10^m gives A275358(m). - _Andrey Zabolotskiy_, Aug 16 2016
%C A175066 Numbers n such that a(n) is nonprime are 1, 26, 110, ... - _Altug Alkan_, Aug 22 2016
%F A175066 If A117453(n) = m^k with k maximal, then a(n) = tau(k) - 1. - _Charlie Neder_, Mar 02 2019
%e A175066 For n = 12, A117453(12) = 4096 and a(12)=5 since there are 5 ways to write 4096 as m^k: 64^2 = 16^3 = 8^4 = 4^6 = 2^12.
%e A175066 729=27^2=9^3=3^6 and 1024=32^2=4^5=2^10 yield a(8)=a(9)=3. - _R. J. Mathar_, Jan 24 2010
%o A175066 (PARI) lista(nn) = {print1(1, ", "); for (i=2, nn, if (po = ispower(i), np = sum(j=2, po, ispower(i, j)); if (np>1, print1(np, ", "));););} \\ _Michel Marcus_, Mar 20 2013
%o A175066 (Python)
%o A175066 from math import gcd
%o A175066 from sympy import mobius, integer_nthroot, factorint, divisor_count, primerange
%o A175066 def A175066(n):
%o A175066     if n == 1: return 1
%o A175066     def bisection(f,kmin=0,kmax=1):
%o A175066         while f(kmax) > kmax: kmax <<= 1
%o A175066         while kmax-kmin > 1:
%o A175066             kmid = kmax+kmin>>1
%o A175066             if f(kmid) <= kmid:
%o A175066                 kmax = kmid
%o A175066             else:
%o A175066                 kmin = kmid
%o A175066         return kmax
%o A175066     def f(x): return int(n+sum(mobius(k)*(integer_nthroot(x,k)[0]-1+sum(integer_nthroot(x,p*k)[0]-1 for p in primerange((x//k).bit_length()))) for k in range(1,x.bit_length())))
%o A175066     return divisor_count(gcd(*factorint(bisection(f,n,n)).values()))-1 # _Chai Wah Wu_, Nov 24 2024
%Y A175066 Cf. A117453.
%K A175066 nonn
%O A175066 1,2
%A A175066 _Jaroslav Krizek_, Jan 23 2010
%E A175066 Corrected and extended by _R. J. Mathar_, Jan 24 2010