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
KEYWORD
nonn
AUTHOR
Leroy Quet, Oct 30 2007
EXTENSIONS
Extended by Ray Chandler and Stefan Steinerberger, Nov 01 2007
STATUS
approved