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A007438 Moebius transform of triangular numbers.
(Formerly M1339)
5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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

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

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-Fran├žois Alcover, Apr 17 2014 *)

CROSSREFS

Cf. A000217.

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

Sequence in context: A265793 A126338 A193718 * A230301 A031457 A044990

Adjacent sequences:  A007435 A007436 A007437 * A007439 A007440 A007441

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Third comment corrected by Mats Granvik, Apr 24 2010

STATUS

approved

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Last modified August 20 13:04 EDT 2017. Contains 290835 sequences.