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A161700
a(n) is the sum of the elements on the antidiagonal of the difference table of the divisors of n.
33
1, 3, 5, 7, 9, 13, 13, 15, 19, 17, 21, 28, 25, 21, 41, 31, 33, 59, 37, 21, 53, 29, 45, 39, 61, 33, 65, 49, 57, 171, 61, 63, 77, 41, 117, 61, 73, 45, 89, -57, 81, 309, 85, 105, 167, 53, 93, -80, 127, 61, 113, 133, 105, 321, 173, 183, 125, 65, 117, -1039, 121, 69, 155, 127, 201, 333, 133, 189, 149, -69, 141, 117, 145, 81, 317, 217, 269
OFFSET
1,2
COMMENTS
a(p^k) = p^(k+1) - (p-1)^(k+1) if p is prime. - Robert Israel, May 18 2016
LINKS
Eric Weisstein's World of Mathematics, Divisor
Eric Weisstein's World of Mathematics, Finite Difference
Reinhard Zumkeller, Enumerations of Divisors
FORMULA
a(n) = EDP(n,tau(n)) with tau = A000005 and EDP(n,x) = interpolating polynomial for the divisors of n.
EDP(n,A000005(n) - 1) = n;
EDP(n,1) = A020639(n);
EDP(n,0) = 1;
EDP(n,k) = A027750(A006218(n-1)+k+1), 0<=k < A000005(n).
EXAMPLE
n=12: A000005(12)=6;
EDP(12,x) = (x^5 - 5*x^4 + 5*x^3 + 5*x^2 + 114*x + 120)/120 = A161701(x) is the interpolating polynomial for {(0,1),(1,2),(2,3),(3,4),(4,6),(5,12)},
{EDP(12,x): 0<=x<6} = {1, 2, 3, 4, 6, 12} = divisors of 12,
a(12) = EDP(12,6) = 28.
From Peter Luschny, May 18 2016: (Start)
a(40) = -57 because the sum of the elements on the antidiagonal of DTD(40) is -57.
The DTD(40) is:
[ 1 2 4 5 8 10 20 40]
[ 1 2 1 3 2 10 20 0]
[ 1 -1 2 -1 8 10 0 0]
[ -2 3 -3 9 2 0 0 0]
[ 5 -6 12 -7 0 0 0 0]
[ -11 18 -19 0 0 0 0 0]
[ 29 -37 0 0 0 0 0 0]
[ -66 0 0 0 0 0 0 0]
(End)
MAPLE
f:= proc(n)
local D, nD;
D:= sort(convert(numtheory:-divisors(n), list));
nD:= nops(D);
CurveFitting:-PolynomialInterpolation([$0..nD-1], D, nD)
end proc:
map(f, [$1..100]); # Robert Israel, May 18 2016
MATHEMATICA
a[n_] := (d = Divisors[n]; t = Table[Differences[d, k], {k, 0, lg = Length[d]}]; Sum[t[[lg - k + 1, k]], {k, 1, lg}]);
Array[a, 77] (* Jean-François Alcover, Jan 25 2018 *)
PROG
(SageMath)
def A161700(n):
D = divisors(n)
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 sum(T[k, len(D)-k-1] for k in range(len(D)))
print([A161700(n) for n in range(1, 78)]) # Peter Luschny, May 18 2016
KEYWORD
sign
AUTHOR
Reinhard Zumkeller, Jun 17 2009, Jun 20 2009
EXTENSIONS
New name from Peter Luschny, May 18 2016
STATUS
approved