A133725 a(n) = Sum_{d|n} mu(n/d)*d*(3*d - 1)/2.

1, 4, 11, 17, 34, 35, 69, 70, 105, 106, 175, 142, 246, 213, 284, 284, 424, 321, 531, 428, 570, 535, 781, 572, 890, 750, 963, 858, 1246, 860, 1425, 1144, 1430, 1288, 1716, 1290, 2034, 1611, 2004, 1720, 2500, 1722, 2751, 2150, 2580, 2365, 3289, 2296, 3507, 2690

Offset: 1

Author

Gary W. Adamson, Sep 21 2007

Keywords

• nonn

Comments

Previous name was: A054525 * A000326.

Möbius transform of the pentagonal numbers.

Examples

			a(4) = 17 = (0, -1, 0, 1) dot (1, 5, 12, 22) = (0, -5, 0, 22).
		

Crossrefs

Cf. A000010, A000326, A007434, A007438, A008683 (mu), A054525.

Programs

Maple

read("transforms") : A000326 := proc(n) n*(3*n-1)/2 ; end: a000326 := [seq(A000326(n),n=1..300)] ; a133725 := MOBIUS(a000326) ; for i from 1 to nops(a133725) do printf("%d,",op(i,a133725)) ; od: # R. J. Mathar, Jan 19 2009

Mathematica

a[n_] := DivisorSum[n, #*(3*#-1) * MoebiusMu[n/#] &] / 2; Array[a, 50] (* Amiram Eldar, May 29 2025 *)

PARI

a(n) = sumdiv(n, d, d*(3*d-1) * moebius(n/d)) / 2; \\ Amiram Eldar, May 29 2025

Formula

G.f.: Sum_{k>=1} mu(k) * x^k * (1 + 2*x^k) / (1 - x^k)^3. - Ilya Gutkovskiy, Sep 17 2021

a(n) = (3*A007434(n) - A000010(n))/2. - Amiram Eldar, Jun 04 2025

Extensions

More terms from R. J. Mathar, Jan 19 2009

New name from Ilya Gutkovskiy, Sep 17 2021