login
a(n) is the sum of the first column of the difference table of the divisors of n.
3

%I #22 Mar 07 2020 08:51:59

%S 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,

%T 29,38,31,6,17,4,31,-5,37,4,19,-42,41,76,43,15,27,4,47,-66,43,-4,23,

%U 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

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

%C 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

%H Reinhard Zumkeller, <a href="/A161857/b161857.txt">Table of n, a(n) for n = 1..1000</a>

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

%e The DTD of 65 is:

%e [ 1 5 13 65]

%e [ 4 8 52]

%e [ 4 44]

%e [ 40]

%e sigma(65) = 1 + 5 + 13 + 65 = 84.

%e a(65) = 1 + 4 + 4 + 40 = 49.

%t 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 *)

%o (Sage)

%o def A161857(n):

%o D = divisors(n)

%o T = matrix(ZZ, len(D))

%o for (m, d) in enumerate(D):

%o T[0, m] = d

%o for k in range(m-1, -1, -1) :

%o T[m-k, k] = T[m-k-1, k+1] - T[m-k-1, k]

%o return sum(T.column(0))

%o print([A161857(n) for n in range(1,78)]) # _Peter Luschny_, May 18 2016

%Y Row sums of A161856.

%Y Cf. A000005, A000203, A006218.

%K sign

%O 1,2

%A _Reinhard Zumkeller_, Jun 20 2009

%E New name from _Peter Luschny_, May 18 2016