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 A015631 Number of ordered triples of integers from [ 1..n ] with no global factor. 12
 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 a(n) = n*(n+1)*(n+2)/6 - Sum_{j=2..n} a(floor(n/j)) = A000292(n) - Sum_{j=2..n} a(floor(n/j)). - Chai Wah Wu, Mar 30 2021 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 (Python) from functools import lru_cache @lru_cache(maxsize=None) def A015631(n):     if n == 0:         return 0     c, j = 1, 2     k1 = n//j     while k1 > 1:         j2 = n//k1 + 1         c += (j2-j)*A015631(k1)         j, k1 = j2, n//j2     return n*(n-1)*(n+4)//6-c+j # Chai Wah Wu, Mar 30 2021 (PARI) a(n) = sum(k=1, n, sumdiv(k, d, moebius(k/d)*binomial(d+1, 2))); \\ Seiichi Manyama, Jun 12 2021 (PARI) a(n) = binomial(n+2, 3)-sum(k=2, n, a(n\k)); \\ Seiichi Manyama, Jun 12 2021 (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k/(1-x^k)^3)/(1-x)) \\ Seiichi Manyama, Jun 12 2021 CROSSREFS Column k=3 of A177976. Cf. A000292, A002088, A015616, A015634, A015650, A134543. Sequence in context: A294423 A294426 A097589 * A116686 A317252 A135350 Adjacent sequences:  A015628 A015629 A015630 * A015632 A015633 A015634 KEYWORD nonn,changed AUTHOR STATUS approved

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Last modified June 22 10:56 EDT 2021. Contains 345375 sequences. (Running on oeis4.)