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

1,2

Links

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

Formula

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

Example

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: (Starts)
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)

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A083111 A354112 A345270 * A157020 A180253 A264786

Adjacent sequences: A132062 A132063 A132064 * A132066 A132067 A132068

Keyword

nonn

Author

Leroy Quet, Oct 30 2007

Extensions

Extended by Ray Chandler and Stefan Steinerberger, Nov 01 2007

Status

approved