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A196439 a(n) = the sum of numbers k <= n such that GCQ_A(n, k) = LCQ_A(n, k) = 0 (see definition in comments). 8
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

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

Antti Karttunen, Table of n, a(n) for n = 1..65537

FORMULA

a(n) = A000217(n) - A196440(n).

EXAMPLE

For n = 6, a(6) = 10 because there are 4 cases k (k = 1, 2, 3, 4) with GCQ_A(6, k) = 0:

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.

Also there are 4 same cases k 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.

PROG

(PARI)

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.

A196440(n) = sum(k=1, n, (2<=GCQ_A(n, k))*k);

A196439(n) = (((n*(n+1))/2) - A196440(n)); \\ Antti Karttunen, Jun 12 2018

CROSSREFS

Cf. A196437, A196438, A196440, A196441, A196442, A196443, A196444.

Sequence in context: A181788 A112163 A058587 * A184389 A163163 A323349

Adjacent sequences:  A196436 A196437 A196438 * A196440 A196441 A196442

KEYWORD

nonn

AUTHOR

Jaroslav Krizek, Nov 26 2011

EXTENSIONS

More terms from Antti Karttunen, Jun 12 2018

STATUS

approved

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Last modified September 24 04:19 EDT 2020. Contains 337317 sequences. (Running on oeis4.)