%I #28 Jul 27 2023 08:20:15
%S 0,0,0,0,1,0,1,0,1,0,2,0,3,1,1,0,2,0,3,1,3,0,3,0,4,1,3,0,2,0,3,3,5,1,
%T 1,0,7,3,3,0,4,0,5,2,5,0,4,0,2,4,6,0,3,1,5,5,7,0,4,0,9,3,2,3,4,0,6,7,
%U 4,0,3,0,10,2,7,1,5,0,5,1,11,0,3,3,11,5,3,0,2,1,8,7,11,4,4,0,3,3,3,0,5,0,7,2
%N The difference between floor(log_2(.)) of and the number of prime factors in A156552(n) (when counted with multiplicity).
%H Antti Karttunen, <a href="/A342657/b342657.txt">Table of n, a(n) for n = 2..10000 (based on Hans Havermann's factorization of A156552)</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F a(n) = A339893(A156552(n)) = A339893(A322993(n)) = A000523(A156552(n)) - A001222(A156552(n)).
%F a(n) = (A252464(n)-A342655(n))-1 = (A325134(n)-A342655(n)) - 2.
%F a(p) = a(p^2) = 0 for all primes p. (Second part added Jul 27 2023)
%F a(A003961(n)) = a(2*A246277(n)) = a(n).
%o (PARI)
%o A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res};
%o A342657(n) = { my(u=A156552(n)); (#binary(u)-bigomega(u))-1; };
%Y Cf. A000430 (positions of 0's), A000523, A001222, A003961, A061395, A070939, A156552, A246277, A322993, A325134, A339893, A342655, A342656.
%Y Cf. also A339895.
%K nonn
%O 2,11
%A _Antti Karttunen_, Mar 18 2021