cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A127415 a(n) = Sum_{1<=k<=n, gcd(k,n)=1}, A000217(k).

Original entry on oeis.org

1, 1, 4, 7, 20, 16, 56, 50, 93, 80, 220, 110, 364, 224, 340, 372, 816, 354, 1140, 580, 966, 880, 2024, 820, 2200, 1456, 2304, 1666, 4060, 1240, 4960, 2856, 3850, 3264, 5180, 2706, 8436, 4560, 6396, 4440, 11480, 3612, 13244, 6710, 8400, 8096, 17296, 6344, 17297, 8600
Offset: 1

Views

Author

Gary W. Adamson, Jan 13 2007

Keywords

Comments

From Wolfdieter Lang, Jun 14 2011: (Start)
Such sums are over a reduced residue system modulo n. See the Apostol reference, p. 133, for the definition or the wikipedia link given under A189918.
This sum over triangular numbers can be found using the results given in exercise 16 of the Apostol reference on p. 48, together with the definition of phi_1(n) and phi_2(n) from the exercise 15.
The result for n >= 2 coincides with the formula given below, using Product_{p|n} (1 - p) = mu(rad(n))*rad(n)*phi(n)/n, with the definitions given there.
(End)

Examples

			a(6) = 16 since the relative primes of 6 are 1 and 5 and (1 + 15) = 16.
a(6) = (6/(3!*2))*(15*6 + 1*6)*(1/2)*(2/3)= 16.

References

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

Links

Crossrefs

Cf. A000217, A023896, A076512/A109395, A189918.

Programs

  • Mathematica
    rad[n_] := Times @@ (FactorInteger[n][[ All, 1]]); a[n_] := (n/(3!*2))*((2*n+3)*n + MoebiusMu[ rad[n]]*rad[n])*(EulerPhi[n] / n); a[1] = 1; Table[ a[n], {n, 1, 33}] (* Jean-François Alcover, Oct 03 2011 *)
  • PARI
    a(n)=if(n<3,return(1));my(s=factor(n)[,1]); s=prod(i=1,#s,s[i]); (n/12)*((2*n+3)*n + moebius(s)*s)*(eulerphi(n)/n) \\ Charles R Greathouse IV, May 17 2011
  • PARI
    a(n) = sum(k=1, n, if (gcd(n,k)==1, k*(k+1)/2)); \\ Michel Marcus, Feb 01 2016

Formula

M * V where M = A054521 is an infinite lower triangular matrix and V = A000217: (1, 3, 6, 10, ...).
From Wolfdieter Lang, May 17 2011: (Start)
a(n) = (n/(3!*2))*((2*n+3)*n + mu(rad(n))*rad(n))*(phi(n)/n), n >= 2, with rad(n) the squarefree kernel of n (the largest squarefree number dividing n, see A007947), the Moebius function mu(n)=A008683(n), and the Euler totient function phi(n)= A000010(n).
Note that phi(n)/n = A076512(n)/A109395(n) = phi(rad(n))/rad(n).
Proof via inclusion-exclusion.
(End)

Extensions

More terms and formula from Wolfdieter Lang, May 17 2011
More terms from Michel Marcus, Feb 01 2016

A192000 Sum of binomial numbers A000332(k+3), with k in the reduced residue system modulo n.

Original entry on oeis.org

0, 1, 6, 16, 56, 71, 252, 296, 651, 721, 2002, 1282, 4368, 3402, 5782, 6672, 15504, 7947, 26334, 15702, 28868, 28457, 65780, 30212, 85580, 63063, 103284, 81452, 201376, 66102, 278256, 174624, 255794, 228684, 383166, 206838, 658008, 391419, 576394, 413244, 1086008
Offset: 1

Views

Author

Wolfdieter Lang, Jun 22 2011

Keywords

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.
This is the m=4 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, 2 and 3 are A000010, A023896, A127415, and A189918, respectively, where in the m=1 and 2 cases the offset for n=1 should be taken as 0 (not 1).

Examples

			a(6) = A000332(4) + A000292(8)= 1 + 70 = 71.
a(6) = (6/6!)*(6*3666*(1/3) + 5*137*2 - 182) = 71.
a(12) = A000332(4) + A000332(8) + A000332(10) + A000332(14) = 1 + 70 + 210 + 1001 = 1282.
a(12) = (12/6!)*(12*18258*(1/3) + 5*407*2 - 182) = 1282.

Crossrefs

Cf. A189918, A023896, A127415, A189922.

Programs

  • PARI
    a(n) = sum(k=0, n-1, if (gcd(n,k) == 1, binomial(k+3, 4))); \\ Michel Marcus, Feb 01 2016

Formula

a(n) = sum(A000332(k+3), 0<=k<=n-1, gcd(k,n)=1), n>=1.
a(n) = (n/6!)*(n*(6*n^3+45*n^2+110*n+90)*P(1,n) + 5*(2*n^2+9*n+11)*P(-1,n) - P(-3,n)), n>=2, with P(k,n):= J(k,n)/n^k, where J(k,n) is the Jordan function (see A000010, A007434, A059376 - A059378, A069091 - A069095).

Extensions

More terms from Michel Marcus, Feb 01 2016
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