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%I #10 Jun 12 2018 21:15:05
%S 1,3,3,6,3,10,3,6,7,6,3,15,3,6,7,6,3,10,3,12,7,6,3,15,3,6,7,6,3,10,3,
%T 6,7,6,3,15,3,6,7,12,3,10,3,6,7,6,3,15,3,6,7,6,3,10,3,6,7,6,3,28,3,6,
%U 7,6,3,10,3,6,7,6,3,15,3,6,7,6,3,10,3,12,7,6,3,15,3,6,7,6,3,10,3,6,7,6,3,15,3,6,7,12,3,10,3,6,7
%N a(n) = the sum of numbers k <= n such that GCQ_A(n, k) = LCQ_A(n, k) = 0 (see definition in comments).
%C Definition of GCQ_A: The greatest common non-divisor of type A (GCQ_A) of two positive integers a and b (a<=b) is the largest positive non-divisor q of numbers a and b such that 1<=q<=a common to a and b; GCQ_A(a, b) = 0 if no such c exists.
%C GCQ_A(1, b) = GCQ_A(2, b) = 0 for b >=1. GCQ_A(a, b) = 0 or >= 2.
%C Definition of LCQ_A: The least common non-divisor of type A (LCQ_A) of two positive integers a and b (a<=b) is the least positive non-divisor q of numbers a and b such that 1<=q<=a common to a and b; LCQ_A(a, b) = 0 if no such c exists.
%C LCQ_A(1, b) = LCQ_A(2, b) = 0 for b >=1. LCQ_A(a, b) = 0 or >= 2.
%H Antti Karttunen, <a href="/A196439/b196439.txt">Table of n, a(n) for n = 1..65537</a>
%F a(n) = A000217(n) - A196440(n).
%e For n = 6, a(6) = 10 because there are 4 cases k (k = 1, 2, 3, 4) with GCQ_A(6, k) = 0:
%e GCQ_A(6, 1) = 0, GCQ_A(6, 2) = 0, GCQ_A(6, 3) = 0, GCQ_A(6, 4) = 0, GCQ_A(6, 5) = 4, GCQ_A(6, 6) = 5. Sum of such numbers k is 10.
%e Also there are 4 same cases k with LCQ_A(6, k) = 0:
%e LCQ_A(6, 1) = 0, LCQ_A(6, 2) = 0, LCQ_A(6, 3) = 0, LCQ_A(6, 4) = 0, LCQ_A(6, 5) = 4, LCQ_A(6, 6) = 4.
%o (PARI)
%o GCQ_A(a, b) = { forstep(m=min(a, b)-1, 2, -1, if(a%m && b%m, return(m))); 0; }; \\ From PARI-program in A196438.
%o A196440(n) = sum(k=1,n,(2<=GCQ_A(n,k))*k);
%o A196439(n) = (((n*(n+1))/2) - A196440(n)); \\ _Antti Karttunen_, Jun 12 2018
%Y Cf. A196437, A196438, A196440, A196441, A196442, A196443, A196444.
%K nonn
%O 1,2
%A _Jaroslav Krizek_, Nov 26 2011
%E More terms from _Antti Karttunen_, Jun 12 2018
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