A132065 a(n) = Sum_{k=1 to d(n)} C(d(n)-1, k-1) d_k, where d(n) is the number of divisors of n and d_k is the k-th divisor of n.

1, 3, 4, 9, 6, 22, 8, 27, 16, 32, 12, 123, 14, 42, 40, 81, 18, 164, 20, 171, 52, 62, 24, 704, 36, 72, 64, 219, 30, 808, 32, 243, 76, 92, 72, 1765, 38, 102, 88, 944, 42, 1016, 44, 315, 276, 122, 48, 4075, 64, 336, 112, 363, 54, 1224, 104, 1170, 124, 152, 60, 17815, 62

Offset: 1

Author

Leroy Quet, Oct 30 2007

Keywords

• nonn

Examples

			Since the divisors of 12 are 1,2,3,4,6,12 and since row (d(12)-1) of Pascal's triangle is 1,5,10,10,5,1, a(12) = 1*1 + 5*2 + 10*3 + 10*4 + 5*6 + 1*12 = 123.
From _Peter Luschny_, May 18 2016: (Start)
Also the lower vertex of the accumulation triangle of the divisors of n.
For instance a(39) = 88 because the lower vertex of ATD(39) = 88. ATD(39) is:
  [ 39  13  3  1]
  [ 52  16  4]
  [ 68  20]
  [ 88]
(End)
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007318 (Pascal's triangle), A027750 (divisors of n).

Programs

Mathematica

f[n_] := Block[{d, l, k},d = Divisors[n];l = Length[d];Sum[ Binomial[l - 1, k - 1]*d[[k]], {k, l}]];Array[f, 100] (* Ray Chandler, Oct 31 2007 *)
Table[Sum[Binomial[Length[Divisors[n]] - 1, k - 1]*Divisors[n][[k]], {k, 1, Length[Divisors[n]]}], {n, 1, 70}] (* Stefan Steinerberger, Oct 31 2007 *)

PARI

a(n) = {d = divisors(n); sum(i=1, #d, d[i]*binomial(#d-1, i-1));} \\ Michel Marcus, Sep 13 2014

Sage

def A132065(n):
    D = divisors(n)[::-1]
    T = matrix(ZZ, len(D))
    for (m, d) in enumerate(D):
        T[0, m] = d
        for k in range(m-1, -1, -1) :
            T[m-k, k] = T[m-k-1, k+1] + T[m-k-1, k]
    return T[len(D)-1,0]
print([A132065(n) for n in range(1,62)]) # Peter Luschny, May 18 2016

Formula

a(p) = p+1, for p prime. - Michel Marcus, Sep 13 2014

Extensions

Extended by Ray Chandler and Stefan Steinerberger, Nov 01 2007