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A286462 Compound filter (3-adic valuation & the length of rightmost run of 1's in base-2): a(n) = P(A051064(n), A089309(n)), where P(n,k) is sequence A000027 used as a pairing function.

%I A286462 #13 Feb 16 2025 08:33:45
%S A286462 1,1,5,1,1,5,4,1,6,1,2,5,1,4,12,1,1,6,2,1,3,2,4,5,1,1,14,4,1,12,11,1,
%T A286462 3,1,2,6,1,2,8,1,1,3,2,2,6,4,7,5,1,1,5,1,1,14,4,4,3,1,2,12,1,11,31,1,
%U A286462 1,3,2,1,3,2,4,6,1,1,5,2,1,8,7,1,15,1,2,3,1,2,8,2,1,6,2,4,3,7,11,5,1,1,9,1,1,5,4,1,3,1,2,14,1,4,12,4,1,3,2,1
%N A286462 Compound filter (3-adic valuation & the length of rightmost run of 1's in base-2): a(n) = P(A051064(n), A089309(n)), where P(n,k) is sequence A000027 used as a pairing function.
%H A286462 Antti Karttunen, <a href="/A286462/b286462.txt">Table of n, a(n) for n = 1..10000</a>
%H A286462 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%F A286462 a(n) = (1/2)*(2 + ((A051064(n)+A089309(n))^2) - A051064(n) - 3*A089309(n)).
%o A286462 (PARI)
%o A286462 A051064(n) = if(n<1, 0, 1+valuation(n, 3));
%o A286462 A089309(n) = valuation((n/2^valuation(n, 2))+1, 2); \\ After _Ralf Stephan_
%o A286462 A286462(n) = (1/2)*(2 + ((A051064(n)+A089309(n))^2) - A051064(n) - 3*A089309(n));
%o A286462 for(n=1, 10000, write("b286462.txt", n, " ", A286462(n)));
%o A286462 (Scheme) (define (A286462 n) (* (/ 1 2) (+ (expt (+ (A051064 n) (A089309 n)) 2) (- (A051064 n)) (- (* 3 (A089309 n))) 2)))
%o A286462 (Python)
%o A286462 from sympy import divisors, divisor_count, mobius
%o A286462 def a051064(n): return -sum([mobius(3*d)*divisor_count(n/d) for d in divisors(n)])
%o A286462 def v(n): return bin(n)[2:][::-1].index("1")
%o A286462 def a089309(n): return  0 if n==0 else v(n/2**v(n) + 1)
%o A286462 def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
%o A286462 def a(n): return T(a051064(n), a089309(n)) # _Indranil Ghosh_, May 11 2017
%Y A286462 Cf. A000027, A051064, A089309, A286362, A286463, A286464.
%K A286462 nonn
%O A286462 1,3
%A A286462 _Antti Karttunen_, May 10 2017