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A196438
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a(n) is the number of integers k <= n such that GCQ_A(n, k) >= 2 (see definition in comments).
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8
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0, 0, 1, 1, 3, 2, 5, 5, 6, 7, 9, 7, 11, 11, 12, 13, 15, 14, 17, 16, 18, 19, 21, 19, 23, 23, 24, 25, 27, 26, 29, 29, 30, 31, 33, 31, 35, 35, 36, 36, 39, 38, 41, 41, 42, 43, 45, 43, 47, 47
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OFFSET
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1,5
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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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.
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PROG
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(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])
(PARI) GCQ_A(a, b)=forstep(m=min(a, b)-1, 2, -1, if(a%m && b%m, return(m))); 0
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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