cp's OEIS Frontend

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

%I A161857 #24 Aug 04 2025 17:37:35
%S A161857 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 A161857 29,38,31,6,17,4,31,-5,37,4,19,-42,41,76,43,15,27,4,47,-66,43,-4,23,
%U A161857 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 A161857 a(n) is the sum of the first column of the difference table of the divisors of n.
%C A161857 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 A161857 Reinhard Zumkeller, <a href="/A161857/b161857.txt">Table of n, a(n) for n = 1..1000</a>
%F A161857 a(n) = SUM(A161856(A006218(n-1)+i): 1<=i<=A000005(n)), n>1.
%e A161857 The DTD of 65 is:
%e A161857 [  1   5  13  65]
%e A161857 [  4   8  52]
%e A161857 [  4  44]
%e A161857 [ 40]
%e A161857 sigma(65) = 1 + 5 + 13 + 65 = 84.
%e A161857 a(65) = 1 + 4 + 4 + 40 = 49.
%t A161857 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 *)
%t A161857 Table[Total[Table[Differences[Divisors[k],n],{n,0,DivisorSigma[0,k]-1}][[;;,1]]],{k,80}] (* _Harvey P. Dale_, Aug 04 2025 *)
%o A161857 (Sage)
%o A161857 def A161857(n):
%o A161857     D = divisors(n)
%o A161857     T = matrix(ZZ, len(D))
%o A161857     for (m, d) in enumerate(D):
%o A161857         T[0, m] = d
%o A161857         for k in range(m-1, -1, -1) :
%o A161857             T[m-k, k] = T[m-k-1, k+1] - T[m-k-1, k]
%o A161857     return sum(T.column(0))
%o A161857 print([A161857(n) for n in range(1,78)]) # _Peter Luschny_, May 18 2016
%Y A161857 Row sums of A161856.
%Y A161857 Cf. A000005, A000203, A006218.
%K A161857 sign
%O A161857 1,2
%A A161857 _Reinhard Zumkeller_, Jun 20 2009
%E A161857 New name from _Peter Luschny_, May 18 2016