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A324735 Difference between the binary length and the binary weight of A324712. a(n) = 0 if A324712(n) = 0.

%I A324735 #13 Jul 27 2024 15:56:03
%S A324735 0,0,0,1,0,2,0,2,0,1,0,0,0,2,2,4,0,0,0,2,2,2,0,3,1,3,3,2,0,2,0,3,3,3,
%T A324735 4,3,0,2,4,5,0,3,0,3,2,3,0,3,2,2,7,3,0,2,4,3,4,5,0,2,0,5,3,5,4,1,0,5,
%U A324735 3,2,0,3,0,3,3,2,4,1,0,3,5,6,0,1,6,4,6,4,0,1,2,5,5,6,5,4,0,3,4,3,0,4,0,7,3
%N A324735 Difference between the binary length and the binary weight of A324712. a(n) = 0 if A324712(n) = 0.
%C A324735 Number of nonleading 0-bits in A324712(n), with a(n) = 0 if A324712(n) = 0.
%H A324735 Antti Karttunen, <a href="/A324735/b324735.txt">Table of n, a(n) for n = 1..10000</a> (based on Hans Havermann's factorization of A156552)
%H A324735 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A324735 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A324735 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A324735 a(n) = A324728(n) - A324829(n).
%F A324735 If A324712(n) > 0, then a(n) = -1 + A324734(n) + A324724(n).
%F A324735 a(p) = 0 for all primes p.
%o A324735 (PARI)
%o A324735 A324712(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A323243(d)))); (v); }; \\ Needs also code from A323243.
%o A324735 A000523(n) = if( n<1, 0, #binary(n) - 1); \\ From A000523
%o A324735 A324735(n) = { my(k=A324712(n)); if(!k,k,1 + (A000523(k)-hammingweight(k))); };
%Y A324735 Cf. A323243, A324712, A324728, A324733, A324829, A324724, A324734.
%K A324735 nonn
%O A324735 1,6
%A A324735 _Antti Karttunen_, Mar 17 2019