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A189918 Sum of tetrahedral numbers A000292(k), with k in the reduced residue system modulo n. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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).

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

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.

EXAMPLE

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.

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 *)

PROG

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

CROSSREFS

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

Sequence in context: A127864 A055936 A194589 * A318415 A164560 A054854

Adjacent sequences:  A189915 A189916 A189917 * A189919 A189920 A189921

KEYWORD

nonn

AUTHOR

Wolfdieter Lang, May 19 2011

STATUS

approved

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Last modified September 28 14:44 EDT 2022. Contains 357073 sequences. (Running on oeis4.)