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A335427 a(1) = 0; for k >= 2, a(prime(k)) = 0, a(k^2) = 2 * a(k); otherwise a(n) = a(A334870(n)) + 1.

%I A335427 #23 Jun 26 2020 23:44:19
%S A335427 0,1,0,2,0,1,0,3,0,1,0,4,0,1,2,4,0,1,0,6,2,1,0,5,0,1,2,10,0,3,0,5,2,1,
%T A335427 4,2,0,1,2,7,0,3,0,18,4,1,0,6,0,1,2,34,0,3,4,11,2,1,0,8,0,1,8,6,4,3,0,
%U A335427 66,2,5,0,3,0,1,2,130,8,3,0,8,0,1,0,12,4,1,2,19,0,5,8,258,2,1,4,7,0,1,16,2,0,3,0,35,6
%N A335427 a(1) = 0; for k >= 2, a(prime(k)) = 0, a(k^2) = 2 * a(k); otherwise a(n) = a(A334870(n)) + 1.
%H A335427 Antti Karttunen, <a href="/A335427/b335427.txt">Table of n, a(n) for n = 1..16384</a>
%H A335427 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A335427 Alternative definition: (Start)
%F A335427 a(1) = 0, a(2) = 1; otherwise for n = k * m^2, k squarefree:
%F A335427   if m = 1, a(n) = A048675(A052126(k));
%F A335427   if m > 1, a(n) = A048675(k) + 2 * a(m).
%F A335427 (End)
%F A335427 For n = 4 * A122132(k), a(n) = A048675(n).
%F A335427 More generally, a(n) = A048675(n) if and only if n is in A335738.
%F A335427 a(n) = A335426(A225546(n)).
%F A335427 a(A003961(2k+1)) = 2 * a(2k+1).
%F A335427 If n is in A036554, a(n) = a(n/2) + 1; otherwise for n <> 3, a(n) = 2 * a(A019565(k/2) * m^2) - a(m^2), where n = A019565(k) * m^2.
%o A335427 (PARI)
%o A335427 A334870(n) = if(issquare(n),sqrtint(n),my(c=core(n), m=n); forprime(p=2, , if(!(c % p), m/=p; break, m*=p)); (m));
%o A335427 A335427(n) = if(n<=2,n-1, if(isprime(n), 0, if(issquare(n), 2*A335427(sqrtint(n)), 1+A335427(A334870(n)))));
%o A335427 (PARI)
%o A335427 A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
%o A335427 A052126(n) = if(1==n,n,(n/vecmax(factor(n)[, 1])));
%o A335427 A335427(n) = if(n<=2,n-1, if(issquarefree(n), A048675(A052126(n)), my(k=core(n)); A048675(k) + 2*A335427(sqrtint(n/k))));
%Y A335427 A052126, A225546, A334870, A335426 are used in formulas defining this sequence.
%Y A335427 Related fully additive sequence: A048675.
%Y A335427 Cf. A062090 (indices of zeros), A003159 (indices of even values), A036554 (indices of odd values).
%Y A335427 Cf. A005117, A122132, A334872, A335738.
%Y A335427 A003961, A019565 are used to express relationship between terms of this sequence.
%K A335427 nonn
%O A335427 1,4
%A A335427 _Antti Karttunen_ and _Peter Munn_, Jun 15 2020