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

Author

Reinhard Zumkeller, Jun 20 2009

Keywords

• sign

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

Examples

			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.
		

Links

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

Crossrefs

Row sums of A161856.

Cf. A000005, A000203, A006218.

Programs

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 *)
Table[Total[Table[Differences[Divisors[k],n],{n,0,DivisorSigma[0,k]-1}][[;;,1]]],{k,80}] (* Harvey P. Dale, Aug 04 2025 *)

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

Formula

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

Extensions

New name from Peter Luschny, May 18 2016