OFFSET
1,5
COMMENTS
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.
GCQ_A(1, b) = GCQ_A(2, b) = 0 for b >=1. GCQ_A(a, b) = 0 or >= 2.
a(n) is also the number of number k <= n such that LCQ_A(n, k) >= 2.
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.
LCQ_A(1, b) = LCQ_A(2, b) = 0 for b >=1. LCQ_A(a, b) = 0 or >= 2.
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = n - A196437(n).
EXAMPLE
For n = 6, a(6) = 2 because there are 2 cases with GCQ_A(6, k) >= 2:
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.
Also there are 2 cases with LCQ_A(6, k) >= 2:
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.
PROG
(PARI) GCQ_A(a, b)=m = min(a, b); if(m < 3, return(0)); da = Set(divisors(a)); db = Set(divisors(b)); s = Set(vector(m-1, i, i)); s = setminus(s, da); s = setminus(s, db); if(#s==0, 0, s[#s])
a(n) = sum(i=3, n, GCQ_A(i, n)>=2) \\ David A. Corneth, Aug 04 2017
(PARI) GCQ_A(a, b)=forstep(m=min(a, b)-1, 2, -1, if(a%m && b%m, return(m))); 0
a(n) = sum(i=3, n, GCQ_A(i, n)>=2) \\ Charles R Greathouse IV, Aug 26 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Jaroslav Krizek, Nov 26 2011
STATUS
approved