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A196437 a(n) = the number of numbers k <= n such that GCQ_A(n, k) = LCQ_A(n, k) = 0 (see definition in comments). 8

%I #10 Mar 20 2018 19:01:45

%S 1,2,2,3,2,4,2,3,3,3,2,5,2,3,3,3,2,4,2,4,3,3,2,5,2,3,3,3,2,4,2,3,3,3,

%T 2,5,2,3,3,4,2,4,2,3,3,3,2,5,2,3,3,3,2,4,2,3,3,3,2,7,2,3,3,3,2,4,2,3,

%U 3,3,2,5,2,3,3,3,2,4,2,4,3,3,2,5,2,3,3,3,2,4,2,3,3,3,2,5,2,3,3,4,2,4,2,3,3

%N a(n) = the number 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="/A196437/b196437.txt">Table of n, a(n) for n = 1..65537</a>

%F a(n) = n - A196438(n).

%e For n = 6, a(6) = 4 because there are 4 cases 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.

%e Also there are 4 cases with LCQ_A(6, k) = 0: 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; };

%o A196438(n) = sum(i=3, n, GCQ_A(i, n)>=2);

%o A196437(n) = (n - A196438(n)); \\ _Antti Karttunen_, Mar 20 2018, based on _Charles R Greathouse IV_'s Aug 26 2017 PARI-program in A196438.

%Y Cf. A196438, A196439, A196440, A196441, A196442, A196443, A196444.

%K nonn

%O 1,2

%A _Jaroslav Krizek_, Nov 26 2011

%E More terms from _Antti Karttunen_, Mar 20 2018

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Last modified March 28 14:13 EDT 2024. Contains 371254 sequences. (Running on oeis4.)