A186230 -id:A186230 - cp's OEIS frontend

A015617 Number of (unordered) triples of integers from [1,n] with no common factors between pairs.

0, 0, 1, 2, 7, 8, 19, 25, 37, 42, 73, 79, 124, 138, 159, 183, 262, 277, 378, 405, 454, 491, 640, 668, 794, 850, 959, 1016, 1257, 1285, 1562, 1668, 1805, 1905, 2088, 2150, 2545, 2673, 2866, 2968, 3457, 3522, 4063, 4228, 4431, 4620, 5269, 5385, 5936

Offset: 1

Olivier Gérard

nonn

Form the graph with nodes 1..n, joining two nodes by an edge if they are relatively prime; a(n) = number of triangles in this graph. - N. J. A. Sloane, Feb 06 2011. The number of edges in this graph is A015614. - Roberto Bosch Cabrera, Feb 07 2011.

			For n=5, there are a(5)=7 triples: (1,2,3), (1,2,5), (1,3,4), (1,3,5), (1,4,5), (2,3,5) and (3,4,5) out of binomial(5,3) = 10 triples of distinct integers <= 5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000

Subset of A015616 (triples with no common factor) and A015631 (ordered triples with no common factor).

Cf. A185953 (first differences), A186230, Column 3 of triangle A186974.

a[n_] := Select[Subsets[Range[n], {3}], And @@ (GCD @@ # == 1 & /@ Subsets[#, {2}]) &] // Length; a /@ Range[49]
(* Jean-François Alcover, Jul 11 2011 *)

a(n)=sum(a=1,n-2,sum(b=a+1,n-1,sum(c=b+1,n, gcd(a,b)==1 && gcd(a,c)==1 && gcd(b,c)==1))) \\ Charles R Greathouse IV, Apr 28 2015

For large n one can show that a(n) ~ C*binomial(n,3), where C = 0.28674... = A065473. - N. J. A. Sloane, Feb 06 2011.

a(n) = Sum_{r=1..n} Sum_{k=1..r} A186230(r,k). - Alois P. Heinz, Feb 17 2011

Added one example and 2 cross-references. - Olivier Gérard, Feb 06 2011.

A185953 Number of pairwise coprime triples of positive integers with largest element n (i.e., A015617(n) - A015617(n-1)).

Original entry on oeis.org

0, 0, 1, 1, 5, 1, 11, 6, 12, 5, 31, 6, 45, 14, 21, 24, 79, 15, 101, 27, 49, 37, 149, 28, 126, 56, 109, 57, 241, 28, 277, 106, 137, 100, 183, 62, 395, 128, 193, 102, 489, 65, 541, 165, 203, 189, 649, 116, 551, 170, 347, 231, 829, 147, 506, 234, 434, 307, 1027, 119, 1101, 364, 450, 412, 727

Offset: 1

N. J. A. Sloane, Feb 07 2011

Robert Israel, Posting to Sequence Fans Mailing List, Feb 06, 2011

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A015617. Row sums of triangle A186230. Column 3 of triangle A186972.

a[n_] := Sum[Boole[GCD[a, n] == 1 && GCD[b, n] == 1 && GCD[a, b] == 1], {a, 1, n-2}, {b, a+1, n-1}]; Array[a, 100] (* Jean-François Alcover, Mar 05 2019, from PARI *)

a(n)=sum(a=1,n-2,sum(b=a+1,n-1,gcd(a,n)==1&&gcd(b,n)==1&&gcd(a,b)==1)) \\ Charles R Greathouse IV, Apr 28 2015

cp's OEIS Frontend

A015617 Number of (unordered) triples of integers from [1,n] with no common factors between pairs.

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