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

Author

Reinhard Zumkeller, Jun 17 2009, Jun 20 2009

Keywords

• sign

Comments

a(p^k) = p^(k+1) - (p-1)^(k+1) if p is prime. - Robert Israel, May 18 2016

Examples

			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)
		

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

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.

Programs

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 *)

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

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).

Extensions

New name from Peter Luschny, May 18 2016