A161857 - OEIS

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

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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