A161857 a(n) is the sum of the first column of the difference table of the divisors of n.

1, 2, 3, 3, 5, 4, 7, 4, 7, 4, 11, 4, 13, 4, 11, 5, 17, 12, 19, -3, 13, 4, 23, -4, 21, 4, 15, 3, 29, 38, 31, 6, 17, 4, 31, -5, 37, 4, 19, -42, 41, 76, 43, 15, 27, 4, 47, -66, 43, -4, 23, 21, 53, 68, 43, 34, 25, 4, 59, -434, 61, 4, 9, 7, 49, 60, 67, 33, 29, -54, 71, 24, 73, 4, 59, 39, 69

Offset

1,2

Comments

Let DTD(n) denote the difference table of the divisors of n. The sum of the first row of DTD(n) is sigma(n) = A000203(n). a(n) is the sum of the first column of DTD(n). - Peter Luschny, May 18 2016

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Formula

a(n) = SUM(A161856(A006218(n-1)+i): 1<=i<=A000005(n)), n>1.

Example

The DTD of 65 is:
[  1   5  13  65]
[  4   8  52]
[  4  44]
[ 40]
sigma(65) = 1 + 5 + 13 + 65 = 84.
a(65) = 1 + 4 + 4 + 40 = 49.

Mathematica

a[n_] := Module[{dd = Divisors[n]}, If[n==1, 1, Sum[Differences[dd, k][[1]], {k, 0, Length[dd]-1}]]]; Array[a, 100] (* Jean-François Alcover, Jun 17 2019 *)

Prog

(Sage)
def A161857(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.column(0))
print([A161857(n) for n in range(1, 78)]) # Peter Luschny, May 18 2016

Crossrefs

Row sums of A161856.

Cf. A000005, A000203, A006218.

Sequence in context: A143092 A143089 A275314 * A135533 A296075 A318678

Adjacent sequences: A161854 A161855 A161856 * A161858 A161859 A161860

Keyword

sign

Author

Reinhard Zumkeller, Jun 20 2009

Extensions

New name from Peter Luschny, May 18 2016

Status

approved