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A196438 a(n) is the number of integers k <= n such that GCQ_A(n, k) >= 2 (see definition in comments). 8
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 (list; graph; refs; listen; history; text; internal format)
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

Cf. A196437, A196439, A196440, A196441, A196442, A196443, A196444.

Sequence in context: A186929 A223538 A059319 * A019828 A289627 A115207

Adjacent sequences:  A196435 A196436 A196437 * A196439 A196440 A196441

KEYWORD

nonn

AUTHOR

Jaroslav Krizek, Nov 26 2011

STATUS

approved

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Last modified October 1 15:05 EDT 2020. Contains 337443 sequences. (Running on oeis4.)