A117108 Moebius transform of tetrahedral numbers.

1, 3, 9, 16, 34, 43, 83, 100, 155, 182, 285, 292, 454, 473, 636, 696, 968, 929, 1329, 1304, 1678, 1735, 2299, 2136, 2890, 2818, 3489, 3484, 4494, 4052, 5455, 5168, 6250, 6168, 7652, 6988, 9138, 8547, 10196, 9840, 12340, 10954, 14189, 13140, 15380, 14993, 18423

Offset: 1

Author

Steve Butler, Apr 18 2006

Keywords

• nonn

Comments

Partial sums of a(n) give A015634(n).

See also A059358, A116963 (applied to shifted version of tetrahedral numbers), inverse Moebius transform of tetrahedral numbers. - Jonathan Vos Post, Apr 20 2006

Examples

			a(2) = 3 because of the triples (1,1,1), (1,1,2), (1,2,2).
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000292 (tetrahedral numbers), A007438, A008683, A015634 (partial sums), A059358, A116963, A117109, A343544.

Programs

Mathematica

a[n_] := DivisorSum[n, MoebiusMu[n/#]*Binomial[# + 2, 3] &]; Array[a, 50] (* Amiram Eldar, Jun 07 2025 *)

PARI

a(n) = sumdiv(n, d, binomial(d+2, 3)*moebius(n/d)); \\ Michel Marcus, Nov 04 2018

Formula

a(n) = |{(x,y,z) : 1 <= x <= y <= z <= n, gcd(x,y,z,n) = 1}|.

G.f.: Sum_{k>=1} mu(k) * x^k / (1 - x^k)^4. - Ilya Gutkovskiy, Feb 13 2020

Extensions

Offset changed to 1 by Ilya Gutkovskiy, Feb 13 2020