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A334109 a(n) = A329697(A225546(n)).

%I A334109 #36 Jun 11 2020 23:42:54
%S A334109 0,0,0,1,0,0,0,1,2,0,0,1,0,0,0,1,0,2,0,1,0,0,0,1,4,0,2,1,0,0,0,1,0,0,
%T A334109 0,3,0,0,0,1,0,0,0,1,2,0,0,1,8,4,0,1,0,2,0,1,0,0,0,1,0,0,2,2,0,0,0,1,
%U A334109 0,0,0,3,0,0,4,1,0,0,0,1,2,0,0,1,0,0,0,1,0,2,0,1,0,0,0,1,0,8,2,5,0,0,0,1,0
%N A334109 a(n) = A329697(A225546(n)).
%C A334109 Conjecture: Each k >= 0 occurs for the first time at A334110(k) = A019565(k)^2. Note that each k must occur first time on square n, because of the identity a(n) = a(A008833(n)). However, is there any reason to exclude squares with prime exponents > 2 from the candidates? See also comments in A334204.
%H A334109 Antti Karttunen, <a href="/A334109/b334109.txt">Table of n, a(n) for n = 1..10200</a>
%H A334109 Antti Karttunen, <a href="/A334109/a334109.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A334109 Additive with a(prime(i)^j) = A000079(i-1) * A329697(A019565(j)), a(m*n) = a(m)+a(n) if gcd(m,n) = 1.
%F A334109 Alternatively, additive with a(prime(i)^(2^k)) = 2^(i-1) * A329697(prime(k+1)), a(m*n) = a(m)+a(n) if A059895(m,n) = 1. - _Peter Munn_, May 04 2020
%F A334109 a(n) = A329697(A225546(n)) = A329697(A331736(n)).
%F A334109 a(n) = a(A008833(n)).
%F A334109 For all n >= 0, a(A334110(n)) = n, a(A334860(n)) = A334204(n).
%F A334109 a(A331590(m,k)) = a(m) + a(k); a(A003961(n)) = 2*a(n). - _Peter Munn_, Apr 30 2020
%t A334109 Map[-1 + Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # != 2^IntegerExponent[#, 2] &] &, Array[If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 105] ] (* _Michael De Vlieger_, May 26 2020 *)
%o A334109 (PARI)
%o A334109 A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
%o A334109 A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
%o A334109 A334109(n) = { my(f=factor(n),pis=apply(primepi,f[,1]),es=f[,2]); sum(k=1,#f~,(2^(pis[k]-1))*A329697(A019565(es[k]))); };
%Y A334109 Cf. A001248, A005117 (positions of zeros), A334110.
%Y A334109 Cf. A003961, A008833, A019565, A059895, A225546, A329697, A331590, A331736, A334204, A334860.
%Y A334109 Cf. also A334107, A334108.
%K A334109 nonn
%O A334109 1,9
%A A334109 _Antti Karttunen_, Apr 29 2020