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 A161700 a(n) = sum of the elements on the antidiagonal of the difference table of the divisors of n. 33
 1, 3, 5, 7, 9, 13, 13, 15, 19, 17, 21, 28, 25, 21, 41, 31, 33, 59, 37, 21, 53, 29, 45, 39, 61, 33, 65, 49, 57, 171, 61, 63, 77, 41, 117, 61, 73, 45, 89, -57, 81, 309, 85, 105, 167, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(p^k) = p^(k+1) - (p-1)^(k+1) if p is prime. - Robert Israel, May 18 2016 LINKS Robert Israel, Table of n, a(n) for n = 1..10000 Eric Weisstein's World of Mathematics, Divisor Eric Weisstein's World of Mathematics, Finite Difference R. Zumkeller, Enumerations of Divisors FORMULA a(n) = EDP(n,tau(n)) with tau = A000005 and EDP(n,x) = interpolating polynomial for the divisors of n. EDP(n,A000005(n) - 1) = n; EDP(n,1) = A020639(n); EDP(n,0) = 1; EDP(n,k) = A027750(A006218(n-1)+k+1), 0<=k < A000005(n). EXAMPLE n=12: A000005(12)=6; 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)}, {EDP(12,x): 0<=x<6} = {1, 2, 3, 4, 6, 12} = divisors of 12, a(12) = EDP(12,6) = 28. From Peter Luschny, May 18 2016: (Start) a(40) = -57 because the sum of the elements on the antidiagonal of DTD(40) is -57. The DTD(40) is: [   1    2    4   5  8  10  20  40] [   1    2    1   3  2  10  20   0] [   1   -1    2  -1  8  10   0   0] [  -2    3   -3   9  2   0   0   0] [   5   -6   12  -7  0   0   0   0] [ -11   18  -19   0  0   0   0   0] [  29  -37    0   0  0   0   0   0] [ -66    0    0   0  0   0   0   0] (End) MAPLE f:= proc(n) local D, nD; D:= sort(convert(numtheory:-divisors(n), list)); nD:= nops(D); CurveFitting:-PolynomialInterpolation([\$0..nD-1], D, nD) end proc: map(f, [\$1..100]); # Robert Israel, May 18 2016 MATHEMATICA a[n_] := (d = Divisors[n]; t = Table[Differences[d, k], {k, 0, lg = Length[d]}]; Sum[t[[lg - k + 1, k]], {k, 1, lg}]); Array[a, 77] (* Jean-François Alcover, Jan 25 2018 *) PROG (Sage) def A161700(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[k, len(D)-k-1] for k in range(len(D))]) print [A161700(n) for n in range(1, 78)] # Peter Luschny, May 18 2016 CROSSREFS 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. Cf. A161856. Sequence in context: A261033 A145341 A121388 * A245212 A063081 A067031 Adjacent sequences:  A161697 A161698 A161699 * A161701 A161702 A161703 KEYWORD sign AUTHOR Reinhard Zumkeller, Jun 17 2009, Jun 20 2009 EXTENSIONS New name from Peter Luschny, May 18 2016 STATUS approved

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Last modified January 23 19:36 EST 2020. Contains 331175 sequences. (Running on oeis4.)