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A285720 Number of ways to write n as a sum of two unordered squarefree numbers so that their addition in base-2 does not produce carries.

%I A285720 #27 Mar 11 2025 02:55:06
%S A285720 0,0,1,0,0,0,2,0,0,0,1,0,0,0,3,0,0,0,1,0,0,0,3,0,0,0,2,0,0,0,5,0,0,0,
%T A285720 2,0,0,0,4,0,0,0,3,0,0,0,6,0,0,0,1,0,0,0,4,0,0,0,4,0,0,0,11,0,0,0,2,0,
%U A285720 0,0,4,0,0,0,3,0,0,0,7,0,0,0,2,0,0,0,6,0,0,0,3,0,0,0,11,0,0,0,3,0,0,0,7,0,0,0,7,0,0,0,13,0,0,0,3,0,0,0,9,0
%N A285720 Number of ways to write n as a sum of two unordered squarefree numbers so that their addition in base-2 does not produce carries.
%H A285720 Antti Karttunen, <a href="/A285720/b285720.txt">Table of n, a(n) for n = 1..4095</a>
%H A285720 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A285720 a(n) = Sum_{i=1..floor(n/2)} abs(mu(i)*mu(n-i))*[A003987(i,n-i) == n]. (Here [] is Iverson bracket, giving in this case 1 only if (i XOR (n-i)) is equal to n, and 0 otherwise. mu is Moebius mu function, A008683.)
%F A285720 a(n) <= A071068(n).
%F A285720 a(n) <= A088512(n).
%t A285720 Table[Sum[Abs[MoebiusMu[i] MoebiusMu[n - i]] Boole[BitXor[i, n - i] == n], {i, Floor[n/2]}], {n, 120}] (* _Michael De Vlieger_, May 03 2017 *)
%o A285720 (Scheme) (define (A285720 n) (let loop ((k (A013928 n)) (s 0)) (if (or (zero? k) (< (A005117 k) (- n (A005117 k)))) s (loop (- k 1) (+ s (if (and (= 1 (A008966 (- n (A005117 k)))) (zero? (A004198bi (A005117 k) (- n (A005117 k))))) 1 0)))))) ;; Where A004198bi implements bitwise-AND (A004198).
%o A285720 (Python)
%o A285720 from sympy import mobius
%o A285720 def a003987(n, i): return i^(n - i) == n
%o A285720 def a(n): return sum([abs(mobius(i)*mobius(n - i))*(1*a003987(n, i)) for i in range(1, n//2 + 1)])
%o A285720 print([a(n) for n in range(1,121)]) # _Indranil Ghosh_, May 02 2017
%Y A285720 Cf. A003987, A004198, A005117, A008683, A008966, A013928, A071068, A088512.
%K A285720 nonn,base,look
%O A285720 1,7
%A A285720 _Antti Karttunen_, May 02 2017