A196440 a(n) = the sum of numbers k <= n such that GCQ_A(n, k) >= 2 (see definition in comments).

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

Offset: 1

Author

Jaroslav Krizek, Nov 26 2011

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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

Formula

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

Extensions

More terms from Antti Karttunen, Jun 12 2018