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%I #23 Mar 31 2020 11:16:12
%S 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 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 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 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 Perfect powers with first occurrence of h >= 2: 16, 64, 65536, 4096, ... (A175065)
%C 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 Numbers n such that a(n) is nonprime are 1, 26, 110, ... - _Altug Alkan_, Aug 22 2016
%F If A117453(n) = m^k with k maximal, then a(n) = tau(k) - 1. - _Charlie Neder_, Mar 02 2019
%e 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 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 (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
%Y Cf. A117453.
%K nonn
%O 1,2
%A _Jaroslav Krizek_, Jan 23 2010
%E Corrected and extended by _R. J. Mathar_, Jan 24 2010
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