|
|
A015631
|
|
Number of ordered triples of integers from [ 1..n ] with no global factor.
|
|
10
|
|
|
1, 3, 8, 15, 29, 42, 69, 95, 134, 172, 237, 287, 377, 452, 552, 652, 804, 915, 1104, 1252, 1450, 1635, 1910, 2106, 2416, 2674, 3007, 3301, 3735, 4027, 4522, 4914, 5404, 5844, 6432, 6870, 7572, 8121, 8805, 9389, 10249, 10831, 11776, 12506
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Number of integer-sided triangles with at least two sides <= n and sides relatively prime. - Henry Bottomley, Sep 29 2006
|
|
LINKS
|
R. J. Mathar, Table of n, a(n) for n = 1..10000
|
|
FORMULA
|
a(n) = (A071778(n)+3*A018805(n)+2)/6. - Vladeta Jovovic, Dec 01 2004
Partial sums of the Moebius transform of the triangular numbers (A007438). - Steve Butler, Apr 18 2006
a(n) = 2*A123324(n) - A046657(n) for n>1. - Henry Bottomley, Sep 29 2006
Row sums of triangle A134543. - Gary W. Adamson, Oct 31 2007
a(n) ~ n^3 / (6*Zeta(3)). - Vaclav Kotesovec, Jan 31 2019
G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - x^k)^3. - Ilya Gutkovskiy, Feb 14 2020
|
|
EXAMPLE
|
a(4) = 15 because the 15 triples in question are in lexicographic order: [1,1,1], [1,1,2], [1,1,3], [1,1,4], [1,2,2], [1,2,3], [1,2,4], [1,3,3], [1,3,4], [1,4,4], [2,2,3], [2,3,3], [2,3,4], [3,3,4] and [3,4,4]. - Wolfdieter Lang, Apr 04 2013
The a(4) = 15 triangles with at least two sides <= 4 and sides relatively prime (see Henry Bottomley's comment above) are: [1,1,1], [1,2,2], [2,2,3], [1,3,3], [2,3,3], [2,3,4], [3,3,4], [3,3,5], [1,4,4], [2,4,5], [3,4,4], [3,4,5], [3,4,6], [4,4,5], [4,4,7]. - Alois P. Heinz, Feb 14 2020
|
|
MAPLE
|
with(numtheory):
b:= proc(n) option remember;
add(mobius(n/d)*d*(d+1)/2, d=divisors(n))
end:
a:= proc(n) option remember;
b(n) + `if`(n=1, 0, a(n-1))
end:
seq(a(n), n=1..60); # Alois P. Heinz, Feb 09 2011
|
|
MATHEMATICA
|
a[1] = 1; a[n_] := a[n] = Sum[MoebiusMu[n/d]*d*(d+1)/2, {d, Divisors[n]}] + a[n-1]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jan 20 2014, after Maple *)
Accumulate[Table[Sum[MoebiusMu[n/d]*d*(d + 1)/2, {d, Divisors[n]}], {n, 1, 50}]] (* Vaclav Kotesovec, Jan 31 2019 *)
|
|
PROG
|
(MAGMA) [n eq 1 select 1 else Self(n-1)+ &+[MoebiusMu(n div d) *d*(d+1)/2:d in Divisors(n)]:n in [1..50]]; // Marius A. Burtea, Feb 14 2020
|
|
CROSSREFS
|
Cf. A002088, A015616, A015634, A015650, A134543.
Sequence in context: A294423 A294426 A097589 * A116686 A317252 A135350
Adjacent sequences: A015628 A015629 A015630 * A015632 A015633 A015634
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Olivier Gérard
|
|
STATUS
|
approved
|
|
|
|