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A196440 a(n) = the sum of numbers k <= n such that GCQ_A(n, k) >= 2 (see definition in comments). 8
0, 0, 3, 4, 12, 11, 25, 30, 38, 49, 63, 63, 88, 99, 113, 130, 150, 161, 187, 198, 224, 247, 273, 285, 322, 345, 371, 400, 432, 455, 493, 522, 554, 589, 627, 651, 700, 735, 773, 808, 858, 893, 943, 984, 1028, 1075, 1125, 1161, 1222, 1269, 1319, 1372, 1428, 1475, 1537, 1590, 1646, 1705, 1767, 1802, 1888, 1947, 2009, 2074, 2142, 2201, 2275 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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 sum 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

Antti Karttunen, Table of n, a(n) for n = 1..1001

FORMULA

a(n) = A000217(n) - A196439(n).

EXAMPLE

For n = 6, a(6) = 11 because there are 2 cases k (k = 5, 6) 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. Sum of such numbers k is 11.

Also there are 2 same cases k  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) = { 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); \\ Antti Karttunen, Jun 12 2018

CROSSREFS

Cf. A196437, A196438, A196439, A196441, A196442, A196443, A196444.

Sequence in context: A034854 A164982 A081837 * A101404 A067327 A218968

Adjacent sequences:  A196437 A196438 A196439 * A196441 A196442 A196443

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 26 21:46 EDT 2020. Contains 337377 sequences. (Running on oeis4.)