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A253951
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A partial double sum of integers: a(n) = Sum_{x=1..n} Sum_{y=1..n} T(x,y), where T is the matrix product: T = A051731*A127093*Transpose(A054524) and T(n,1)=0 (* stands for matrix multiplication).
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1
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0, 1, 5, 9, 20, 23, 42, 52, 69, 77, 113, 119, 165, 177, 190, 214, 279, 291, 366, 379, 399, 422, 517, 533, 599, 625, 679, 701, 829, 846, 986, 1035, 1069, 1105, 1137, 1164, 1339, 1380, 1417, 1449, 1646, 1674, 1883, 1918, 1955, 2008, 2239, 2274, 2420, 2462, 2515, 2559, 2827, 2874, 2929
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OFFSET
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1,3
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COMMENTS
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The exact relation is: lim_{n->Infinity} log(A003418(k))*n = Sum_{x=1..n} Sum_{y=1..k} T(x,y), where T is the matrix product: T = A051731*A127093*Transpose(A054524) and T(n,1)=0.
Compare a(n) to round(log(A003418)*n)= 0, 1, 5, 10, 20, 25, 42, 54, 70, 78,...
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LINKS
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FORMULA
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a(n) = Sum_{x=1..n} Sum_{y=1..n} T(x,y), where T is the matrix product: T=A051731*A127093*Transpose(A054524) and T(n,1)=0. (* stands for matrix multiplication)
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MAPLE
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with(LinearAlgebra):
N:= 200:
A051731:= Matrix(N, N, (n, k) -> `if`(n mod k = 0, 1, 0), shape=triangular[lower]):
A127093:= Matrix(N, N, (n, k) -> `if`(n mod k = 0, k, 0), shape=triangular[lower]):
A054524T:= Matrix(N, N, (k, n) -> `if`(n mod k = 0, numtheory:-mobius(k), 0), shape=triangular[upper]):
a[1]:= 0:
for n from 2 to N do
a[n]:= a[n-1] + add(T[i, n], i=1..n) + add(T[n, j], j=2..n-1)
od:
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MATHEMATICA
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nn = 55;
Z = Table[ If[ Mod[n, k] == 0, 1, 0], {n, nn}, {k, nn}];
A = Table[ If[ Mod[n, k] == 0, k, 0], {n, nn}, {k, nn}];
B = Table[ If[ Mod[n, k] == 0, MoebiusMu[k], 0], {n, nn}, {k, nn}];
MatrixForm[T = Z.A.Transpose[B]];
T[[All, 1]] = 0;
a = Table[ Total[ T[[1 ;; n, 1 ;; n]], 2], {n, nn}]
(* shows a graph *) Show[ ListLinePlot[a], ListLinePlot[ Accumulate[ MangoldtLambda[ Range[ nn]]]]]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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