A189918 Sum of tetrahedral numbers A000292(k), with k in the reduced residue system modulo n.

0, 1, 5, 11, 35, 36, 126, 130, 264, 260, 715, 406, 1365, 952, 1530, 1716, 3876, 1830, 5985, 3300, 5796, 5500, 12650, 5460, 15075, 10556, 16965, 12810, 31465, 9920, 40920, 24616, 34650, 30192, 49210, 26106, 82251, 46740, 67158, 47320

Offset: 1

Author

Wolfdieter Lang, May 19 2011

Keywords

• nonn

Comments

The reduced residue system modulo n used here is the set of numbers k from the set {0,1,...,n-1} which satisfy gcd(k,n)=1. There are phi(n) = A000010(n) such numbers k. Cf. A038566. See also the Apostol reference p. 133, and the Wikipedia link.

This is the m=3 member of a family of sequences, call them rmnS(m) (reduced mod n sum), with entries rmnS(m;n):=sum(binomial(k+m-1,m),0<=k<=n-1 with gcd(k,n)=1), m>=0, n>=1. Recall gcd(0,n)=n.

The members for m=0, 1, and 2 are A000010(n), A023896(n) and A127415(n), respectively, where in the last two the offset for n=1 should be taken as 0 (not 1).

Examples

			a(6) = A000292(1) + A000292(5)= 1 + 35 = 36.
a(6) = (6*8/4!)*(6*8 + 1*6)*((1/2)*(2/3)) = 36.
a(12) = A000292(1) + A000292(5) + A000292(7) + A000292(11) = 1 + 35 + 84 + +286 = 406.
a(12) = (12*14/4!)*(12*14 + 1*6)*((1/2)*(2/3)) = 406.
		

References

• T. Apostol, Introduction to Analytic Number Theory, Springer, 1986.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Wikipedia, Reduced residue system

Crossrefs

Cf. A000010, A038566, A076512/A109395, A023896, A127415.

Programs

Maple

A000292 := proc(n) binomial(n+2,3) ; end proc:
A189918 := proc(n) local a; a := 0 ; for k from 0 to n-1 do if igcd(k,n) = 1 then a := a+A000292(k); end if; end do: a ; end proc:
seq(A189918(n),n=1..40) ; # R. J. Mathar, Jun 13 2011

Mathematica

a[n_] := Sum[ Boole[GCD[k, n] == 1]*k*(k+1)*(k+2)/6, {k, 0, n-1}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jul 12 2012 *)

PARI

a(n) = sum(k=0, n-1, if (gcd(n,k)==1, k*(k+1)*(k+2)/6)); \\ Michel Marcus, Feb 01 2016

Formula

a(n) = Sum_{k=0..n-1, gcd(k,n)=1 } * A000292(k), n>=1.

a(n) = (n*(n+2)/4!) *{n*(n+2) + mu(rad(n))*rad(n)} *phi(n)/n, n>=2, with rad(n) = A007947(n) the squarefree kernel of n, mu(n)=A008683(n), and phi(n)= A000010(n).

Note that phi(n)/n = A076512(n)/A109395(n) = phi(rad(n))/rad(n).

Proof by principle of inclusion-exclusion.