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%I #32 Apr 23 2020 11:20:33
%S 1,3,5,7,9,13,13,15,19,17,21,28,25,21,41,31,33,59,37,21,53,29,45,39,
%T 61,33,65,49,57,171,61,63,77,41,117,61,73,45,89,-57,81,309,85,105,167,
%U 53,93,-80,127,61,113,133,105,321,173,183,125,65,117,-1039,121,69,155,127,201,333,133,189,149,-69,141,117,145,81,317,217,269
%N a(n) is the sum of the elements on the antidiagonal of the difference table of the divisors of n.
%C a(p^k) = p^(k+1) - (p-1)^(k+1) if p is prime. - _Robert Israel_, May 18 2016
%H Robert Israel, <a href="/A161700/b161700.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Divisor.html">Divisor</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FiniteDifference.html">Finite Difference</a>
%H Reinhard Zumkeller, <a href="/A161700/a161700.txt">Enumerations of Divisors</a>
%F a(n) = EDP(n,tau(n)) with tau = A000005 and EDP(n,x) = interpolating polynomial for the divisors of n.
%F EDP(n,A000005(n) - 1) = n;
%F EDP(n,1) = A020639(n);
%F EDP(n,0) = 1;
%F EDP(n,k) = A027750(A006218(n-1)+k+1), 0<=k < A000005(n).
%e n=12: A000005(12)=6;
%e EDP(12,x) = (x^5 - 5*x^4 + 5*x^3 + 5*x^2 + 114*x + 120)/120 = A161701(x) is the interpolating polynomial for {(0,1),(1,2),(2,3),(3,4),(4,6),(5,12)},
%e {EDP(12,x): 0<=x<6} = {1, 2, 3, 4, 6, 12} = divisors of 12,
%e a(12) = EDP(12,6) = 28.
%e From _Peter Luschny_, May 18 2016: (Start)
%e a(40) = -57 because the sum of the elements on the antidiagonal of DTD(40) is -57.
%e The DTD(40) is:
%e [ 1 2 4 5 8 10 20 40]
%e [ 1 2 1 3 2 10 20 0]
%e [ 1 -1 2 -1 8 10 0 0]
%e [ -2 3 -3 9 2 0 0 0]
%e [ 5 -6 12 -7 0 0 0 0]
%e [ -11 18 -19 0 0 0 0 0]
%e [ 29 -37 0 0 0 0 0 0]
%e [ -66 0 0 0 0 0 0 0]
%e (End)
%p f:= proc(n)
%p local D, nD;
%p D:= sort(convert(numtheory:-divisors(n),list));
%p nD:= nops(D);
%p CurveFitting:-PolynomialInterpolation([$0..nD-1],D, nD)
%p end proc:
%p map(f, [$1..100]); # _Robert Israel_, May 18 2016
%t a[n_] := (d = Divisors[n]; t = Table[Differences[d, k], {k, 0, lg = Length[d]}]; Sum[t[[lg - k + 1, k]], {k, 1, lg}]);
%t Array[a, 77] (* _Jean-François Alcover_, Jan 25 2018 *)
%o (Sage)
%o def A161700(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[k,len(D)-k-1] for k in range(len(D)))
%o print([A161700(n) for n in range(1,78)]) # _Peter Luschny_, May 18 2016
%Y Cf. A000012, A000027, A005408, A000124, A016813, A086514, A016921, A000125, A058331, A002522, A017281, A161701, A017533, A161702, A161703, A000127, A158057, A161704, A161705, A161706, A161707, A161708, A161709, A161710, A080856, A161711, A161712, A161713, A161714, A161715, A128470, A006261.
%Y Cf. A161856.
%K sign
%O 1,2
%A _Reinhard Zumkeller_, Jun 17 2009, Jun 20 2009
%E New name from _Peter Luschny_, May 18 2016
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