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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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..61.

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: A003959 A168341 A083111 * 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

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Last modified June 25 07:53 EDT 2019. Contains 324347 sequences. (Running on oeis4.)