A161700 a(n) is the sum of the elements on the antidiagonal of the difference table of the divisors of n.

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

Robert Israel, Table of n, a(n) for n = 1..10000

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

(Sage)
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

Crossrefs

Cf. A000012, A000027, A005408, A000124, A016813, A086514, A016921, A000125, A058331, A002522, A017281, A161701, A017533, A161702, A161703, A000127, A158057, A161704, A161705, A161706, A161707, A161708, A161709, A161710, A080856, A161711, A161712, A161713, A161714, A161715, A128470, A006261.

Cf. A161856.

Sequence in context: A261033 A145341 A121388 * A245212 A063081 A337431

Adjacent sequences: A161697 A161698 A161699 * A161701 A161702 A161703

Keyword

sign

Author

Reinhard Zumkeller, Jun 17 2009, Jun 20 2009

Extensions

New name from Peter Luschny, May 18 2016

Status

approved