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A253557 a(1) = 0; after which, a(2n) = 1 + a(n), a(2n+1) = a(A268674(2n+1)).

%I A253557 #16 Apr 05 2018 20:33:09
%S A253557 0,1,1,2,1,2,1,3,2,2,1,3,1,2,2,4,1,3,1,3,3,2,1,4,2,2,2,3,1,3,1,5,3,2,
%T A253557 2,4,1,2,2,4,1,4,1,3,4,2,1,5,2,3,3,3,1,3,3,4,3,2,1,4,1,2,2,6,2,4,1,3,
%U A253557 4,3,1,5,1,2,2,3,2,3,1,5,3,2,1,5,3,2,3,4,1,5,3,3,5,2,2,6,1,3,2,4,1,4,1,4,4,2,1,4,1,4,2,5,1
%N A253557 a(1) = 0; after which, a(2n) = 1 + a(n), a(2n+1) = a(A268674(2n+1)).
%C A253557 Consider the binary trees illustrated in A252753 and A252755: If we start from any n, computing successive iterations of A253554 until 1 is reached (i.e., we are traversing level by level towards the root of the tree, starting from that vertex of the tree where n is located), a(n) gives the number of even numbers encountered on the path (i.e., including both 2 and the starting n if it was even).
%C A253557 This is bigomega (A001222) analog for nonstandard factorization based on the sieve of Eratosthenes (A083221). See A302041 for an omega-analog. - _Antti Karttunen_, Mar 31 2018
%H A253557 Antti Karttunen, <a href="/A253557/b253557.txt">Table of n, a(n) for n = 1..8192</a>
%F A253557 a(1) = 0; after which, a(2n) = 1 + a(n), a(2n+1) = a(A268674(2n+1)).
%F A253557 a(n) = A253555(n) - A253556(n).
%F A253557 a(n) = A000120(A252754(n)). [Binary weight of A252754(n).]
%F A253557 Other identities.
%F A253557 For all n >= 0:
%F A253557 a(2^n) = n.
%F A253557 For all n >= 2:
%F A253557 a(n) = A080791(A252756(n)) + 1. [One more than the number of nonleading 0-bits in A252756(n).]
%F A253557 From _Antti Karttunen_, Apr 01 2018: (Start)
%F A253557 a(1) = 0; for n > 1, a(n) = 1 + a(A302042(n)).
%F A253557 a(n) = A001222(A250246(n)).
%F A253557 (End)
%o A253557 (Scheme)
%o A253557 (definec (A253557 n) (cond ((= 1 n) 0) ((odd? n) (A253557 (A250470 n))) (else (+ 1 (A253557 (/ n 2))))))
%Y A253557 Essentially, one more than A253559.
%Y A253557 Primes, A000040, gives the positions of ones.
%Y A253557 Cf. A000079, A000120, A080791, A252753, A252754, A252755, A252756, A253554, A253555, A253556, A253558, A302037, A302041, A302042.
%Y A253557 Differs from A001222 for the first time at n=21, where a(21) = 3, while A001222(21) = 2.
%K A253557 nonn
%O A253557 1,4
%A A253557 _Antti Karttunen_, Jan 12 2015
%E A253557 Definition (formula) corrected by _Antti Karttunen_, Mar 31 2018