A007438 Moebius transform of triangular numbers.

1, 2, 5, 7, 14, 13, 27, 26, 39, 38, 65, 50, 90, 75, 100, 100, 152, 111, 189, 148, 198, 185, 275, 196, 310, 258, 333, 294, 434, 292, 495, 392, 490, 440, 588, 438, 702, 549, 684, 584, 860, 582, 945, 730, 876, 803, 1127, 776, 1197, 910, 1168, 1020, 1430

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

a(n)=|{(x,y):1<=x<=y<=n, gcd(x,y,n)=1}|. E.g. a(4)=7 because of the pairs (1,1), (1,2), (1,3), (1,4), (2,3), (3,3), (3,4). - Steve Butler, Apr 18 2006

Partial sums of a(n) give A015631(n). - Steve Butler, Apr 18 2006

Equals row sums of triangle A159905. - Gary W. Adamson, Apr 25 2009; corrected by Mats Granvik, Apr 24 2010

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000217.

Cf. A159905. - Gary W. Adamson, Apr 25 2009

Programs

Maple

with(numtheory):
a:= proc(n) option remember;
       add(mobius(n/d)*d*(d+1)/2, d=divisors(n))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Feb 09 2011

Mathematica

a[n_] := Sum[MoebiusMu[n/d]*d*(d+1)/2, {d, Divisors[n]}]; Array[a, 60] (* Jean-François Alcover, Apr 17 2014 *)

PARI

a(n) = sumdiv(n, d, moebius(n/d)*d*(d+1)/2); \\ Michel Marcus, Nov 05 2018

Formula

a(n) = (A007434(n)+A000010(n))/2, half the sum of the Mobius transforms of n^2 and n. Dirichlet g.f. (zeta(s-2)+zeta(s-1))/(2*zeta(s)). - R. J. Mathar, Feb 09 2011

G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = x/(1 - x)^3. - Ilya Gutkovskiy, Apr 25 2017